Your FREE ALEKS PPL Math Placement Practice Test 2026 – 220+ Q&A

ALEKS Practice Questions

1. Evaluate the expression 24 - 4 squared / 2 + 6 using the order of operations.
   • 166
   • 22
   • 30
   • 8

   Correct answer: 22

   The value is 22. Order of operations requires exponents first: 4 squared is 16. Then division: 16 divided by 2 is 8. The expression becomes 24 - 8 + 6, and working left to right gives 24 - 8 = 16, then 16 + 6 = 22. Subtracting 8 only after adding 6 to it would change the grouping and give a different, incorrect value.

2. Which calculation correctly applies the order of operations to 2 + 6 / 3 x 4?
   • Divide 6 by 3, multiply by 4, then add 2 to get 10.
   • Add 2 and 6, divide by 3, then multiply by 4 to get about 10.7.
   • Multiply 3 by 4, divide 6 by 12, then add 2 to get 2.5.
   • Add everything in sequence to get 6.

   Correct answer: Divide 6 by 3, multiply by 4, then add 2 to get 10.

   The correct approach divides 6 by 3, multiplies by 4, then adds 2 to get 10. Multiplication and division share the same priority and are performed left to right, so 6 / 3 is done first to get 2, then 2 x 4 gives 8, and finally 2 + 8 equals 10. Multiplying 3 by 4 first violates the left-to-right rule for operations of equal priority, which is why that path produces a wrong result.

3. Find 7/12 + 3/8 and write the answer in lowest terms.
   • 10/20
   • 23/24
   • 5/10
   • 17/24

   Correct answer: 23/24

   The sum is 23/24. The least common denominator of 12 and 8 is 24, so 7/12 becomes 14/24 and 3/8 becomes 9/24. Adding the numerators gives 14 + 9 = 23, producing 23/24, which has no common factor to reduce. The choice 10/20 comes from adding numerators and denominators straight across, which is not how fractions are added.

4. Subtract 2 1/3 - 5/6 and express the result as a mixed number in lowest terms.
   • 1 1/3
   • 2 1/2
   • 1 5/6
   • 1 1/2

   Correct answer: 1 1/2

   The difference is 1 1/2. Rewriting 2 1/3 as an improper fraction gives 7/3, and using a common denominator of 6 turns 7/3 into 14/6 and leaves 5/6 unchanged. Subtracting gives 14/6 - 5/6 = 9/6, which reduces to 3/2, or 1 1/2 as a mixed number. Subtracting only the fraction parts while ignoring the borrowing needed from the whole number would give an incorrect smaller value.

5. What is 4/9 of 6/10, written in lowest terms?
   • 10/19
   • 24/90
   • 4/15
   • 2/3

   Correct answer: 4/15

   The product is 4/15. The word 'of' signals multiplication, so multiply numerators and denominators: 4 x 6 = 24 and 9 x 10 = 90, giving 24/90. Dividing both parts by their greatest common factor 6 reduces 24/90 to 4/15. The choice 24/90 is the same value before reducing, so it is not in lowest terms.

6. A recipe calls for 3/4 cup of flour per batch. How many full batches can be made from 6 cups of flour?
   • 4 1/2
   • 6
   • 8
   • 9

   Correct answer: 8

   The answer is 8 batches. Finding how many 3/4-cup portions fit into 6 cups means dividing 6 by 3/4, which equals multiplying 6 by the reciprocal 4/3. That gives 24/3 = 8, so exactly 8 full batches can be made. Multiplying 6 by 3/4 instead of dividing would wrongly give 4 1/2, but that asks how much flour, not how many batches.

7. Convert the fraction 5/8 to a decimal.
   • 0.58
   • 0.625
   • 0.85
   • 0.16

   Correct answer: 0.625

   The decimal form is 0.625. Dividing the numerator by the denominator, 5 divided by 8 equals 0.625, a terminating decimal. The value 0.58 simply rearranges the original digits rather than performing the division, so it does not represent the same quantity as 5/8.

8. A shirt costs $32 before tax. With an 8% sales tax added, what is the total price?
   • $40.00
   • $32.08
   • $34.56
   • $33.60

   Correct answer: $34.56

   The total price is $34.56. The tax is 8% of $32, which is 0.08 x 32 = $2.56. Adding the tax to the original price gives 32 + 2.56 = $34.56. Treating 8% as $8 and adding it to get $40.00 misreads the percent as a flat dollar amount rather than a fraction of the price.

9. 45 is what percent of 180?
   • 40%
   • 4%
   • 75%
   • 25%

   Correct answer: 25%

   The answer is 25%. To find what percent one number is of another, divide the part by the whole and multiply by 100: 45 divided by 180 equals 0.25, which is 25%. Dividing in the reverse order, 180 by 45, would give 4 and lead to a wrong reading of 400%, so the part must always be divided by the whole.

10. A blueprint uses a scale where 2 inches represents 15 feet. How many feet does a length of 9 inches represent?
    • 67.5
    • 30
    • 120
    • 22.5

    Correct answer: 67.5

    The length represents 67.5 feet. Setting up the proportion 2 inches over 15 feet equal to 9 inches over x feet and cross-multiplying gives 2x = 15 x 9 = 135, so x = 67.5 feet. The value 30 comes from multiplying the 15 feet by the ratio of inches incorrectly, ignoring that 9 inches is 4.5 times the 2-inch unit, which must scale the 15 feet by the same factor.

11. Solve the linear equation 5(x - 3) + 4 = 2x + 7 for x.
    • 6
    • 2
    • 9
    • 4

    Correct answer: 6

    The solution is x = 6. Distributing the 5 gives 5x - 15 + 4 = 2x + 7, which simplifies to 5x - 11 = 2x + 7. Subtracting 2x from both sides and adding 11 gives 3x = 18, so x = 6. Forgetting to distribute the 5 to the -3 inside the parentheses would drop the -15 term and lead to a smaller, incorrect value.

12. A taxi charges a flat fee of $4 plus $2.50 per mile. If a ride costs $26.50, the relationship is 4 + 2.5m = 26.50. How many miles was the ride?
    • 12.2
    • 11
    • 9
    • 9.4

    Correct answer: 9

    The ride was 9 miles. Subtracting the flat fee from both sides gives 2.5m = 22.50, and dividing both sides by 2.5 gives m = 9. Dividing the full $26.50 by 2.5 without first removing the $4 flat fee would overstate the distance, because the flat fee is charged once rather than per mile.

13. Solve the inequality -3x + 5 > 14 and choose the correct solution set.
    • X > -3
    • X < -3
    • X > 3
    • X < 3

    Correct answer: X < -3

    The solution is x < -3. Subtracting 5 from both sides gives -3x > 9, and dividing both sides by -3 requires reversing the inequality symbol, producing x < -3. Keeping the symbol as 'greater than' after dividing by a negative number is the classic error, since dividing an inequality by a negative value always flips the direction of the sign.

14. Which graph on a number line correctly represents the solution to 2x - 1 <= 7?
    • An open circle at 4 shading to the right
    • A closed circle at 4 shading to the left
    • A closed circle at 4 shading to the right
    • An open circle at 4 shading to the left

    Correct answer: A closed circle at 4 shading to the left

    The correct picture is a closed circle at 4 shading to the left. Adding 1 to both sides gives 2x <= 8, then dividing by 2 gives x <= 4, so 4 itself is included and all smaller values satisfy the inequality. A closed (filled) circle marks that 4 is part of the solution, and shading left covers the values less than 4; an open circle would wrongly exclude 4.

15. Solve the absolute value equation |2x - 5| = 9 for all values of x.
    • X = 7 only
    • X = 2 and x = -7
    • X = 7 and x = -2
    • X = -2 only

    Correct answer: X = 7 and x = -2

    The solutions are x = 7 and x = -2. An absolute value equal to 9 means the inside expression equals either 9 or -9, giving two equations: 2x - 5 = 9 yields x = 7, and 2x - 5 = -9 yields x = -2. Solving only the positive case and reporting x = 7 alone misses the second branch, which is why absolute value equations typically produce two solutions.

16. Solve the system of equations: 3x + y = 11 and x - y = 1.
    • X = 3, y = 2
    • X = 2, y = 5
    • X = 4, y = -1
    • X = 1, y = 8

    Correct answer: X = 3, y = 2

    The solution is x = 3 and y = 2. Adding the two equations eliminates y because y and -y cancel, giving 4x = 12, so x = 3. Substituting x = 3 into x - y = 1 gives 3 - y = 1, so y = 2. The ordered pair (3, 2) satisfies both equations, while a pair that works in only one equation cannot be the system's solution.

17. A vendor sells adult tickets for $8 and child tickets for $5. On a day with 200 tickets sold for $1,330 total, the system is a + c = 200 and 8a + 5c = 1330. How many adult tickets were sold?
    • 90
    • 120
    • 80
    • 110

    Correct answer: 110

    The vendor sold 110 adult tickets. Solving a + c = 200 for c gives c = 200 - a, and substituting into 8a + 5c = 1330 gives 8a + 5(200 - a) = 1330, which simplifies to 3a + 1000 = 1330, so 3a = 330 and a = 110. Substituting c instead of solving for the adult count, or mixing up which price goes with which ticket type, would produce a count that fails the revenue equation.

18. Solve the quadratic equation $x^2 - 7x + 12 = 0$ by factoring.
    • X = -3 and x = -4
    • X = 3 and x = 4
    • X = 2 and x = 6
    • X = -2 and x = -6

    Correct answer: X = 3 and x = 4

    The solutions are x = 3 and x = 4. The trinomial factors into (x - 3)(x - 4) because -3 and -4 multiply to +12 and add to -7. Setting each factor equal to zero gives x = 3 and x = 4. Choosing -3 and -4 reverses the signs; those values multiply to +12 but add to -7 only when the factors are (x - 3)(x - 4), so the roots are the positive numbers.

19. Use the quadratic formula to solve $2x^2 + 3x - 2 = 0$.
    • $x = \frac{1}{2}$ and $x = -2$
    • $x = -\frac{1}{2}$ and $x = 2$
    • $x = 2$ and $x = -3$
    • $x = -1$ and $x = 2$

    Correct answer: $x = \frac{1}{2}$ and $x = -2$

    The solutions are $x = \frac{1}{2}$ and $x = -2$. With $a = 2$, $b = 3$, and $c = -2$, the discriminant $b^2 - 4ac$ equals $9 - 4(2)(-2) = 9 + 16 = 25$, whose square root is 5. The formula gives $x = \frac{-3 \pm 5}{4}$, producing $\frac{-3 + 5}{4} = \frac{1}{2}$ and $\frac{-3 - 5}{4} = -2$. Treating the c value as +2 instead of -2 would make the discriminant smaller and yield the wrong roots.

20. For the equation $3x^2 - 5x + 4 = 0$, the quadratic formula uses the discriminant $b^2 - 4ac$. What does the discriminant reveal about the solutions?
    • Two distinct real solutions, because the discriminant is positive
    • Exactly one real solution, because the discriminant is zero
    • No real solutions, because the discriminant is negative
    • Three solutions, because the equation is quadratic

    Correct answer: No real solutions, because the discriminant is negative

    The equation has no real solutions because the discriminant is negative. With $a = 3$, $b = -5$, and $c = 4$, the discriminant $b^2 - 4ac$ equals $25 - 4(3)(4) = 25 - 48 = -23$. A negative value under the square root means no real number satisfies the equation. Concluding two real solutions would require a positive discriminant, which this equation does not have.

21. On the unit circle, what is the value of $\cos(60^\circ)$?
    • $\frac{1}{2}$
    • $\frac{\sqrt{3}}{2}$
    • $\frac{\sqrt{2}}{2}$
    • 0

    Correct answer: $\frac{1}{2}$

    The value is $\frac{1}{2}$. On the unit circle a point is written as $(\cos\theta, \sin\theta)$, and the terminal side of 60 degrees lands at the coordinates $(\frac{1}{2}, \frac{\sqrt{3}}{2})$. The x-coordinate of that point gives the cosine, which is $\frac{1}{2}$. Reporting $\frac{\sqrt{3}}{2}$ mistakes the y-coordinate (the sine of 60 degrees) for the cosine.

22. A right triangle has legs of length 9 and 12. What is the length of the hypotenuse?
    • 21
    • 15
    • $\sqrt{63}$
    • 225

    Correct answer: 15

    The hypotenuse is 15. The Pythagorean theorem states that the square of the hypotenuse equals the sum of the squares of the legs, so $c^2 = 9^2 + 12^2 = 81 + 144 = 225$, and $\sqrt{225}$ is 15. Choosing 225 stops at $c^2$ without taking the square root, and choosing 21 simply adds the legs instead of using the theorem.

23. What is the distance between the points (2, -1) and (7, 11) in the coordinate plane?
    • 13
    • 17
    • $\sqrt{119}$
    • 12

    Correct answer: 13

    The distance is 13. The distance formula takes $\sqrt{(\Delta x)^2 + (\Delta y)^2}$, the root of the sum of the squared differences in the coordinates: the horizontal change is $7 - 2 = 5$ and the vertical change is $11 - (-1) = 12$, giving $\sqrt{5^2 + 12^2}$ = $\sqrt{25 + 144}$ = $\sqrt{169}$ = 13. Adding the differences 5 and 12 to get 17 skips the squaring and square-root steps that the formula requires.

24. What is the midpoint of the segment connecting (-4, 6) and (10, -2)?
    • (7, 4)
    • (-2, 2)
    • (3, 2)
    • (14, -8)

    Correct answer: (3, 2)

    The midpoint is (3, 2). The midpoint formula averages the x-coordinates and the y-coordinates separately: the x-coordinate is $\frac{-4 + 10}{2} = \frac{6}{2} = 3$, and the y-coordinate is $\frac{6 + (-2)}{2} = \frac{4}{2} = 2$. The point (14, -8) results from subtracting the coordinates instead of averaging them, which finds a difference rather than the center of the segment.

25. A cylindrical water tank has a radius of 5 feet and a height of 8 feet. Using the formula $V = \pi r^2 h$, what is its volume in terms of $\pi$?
    • $80\pi$ cubic feet
    • $40\pi$ cubic feet
    • $200\pi$ cubic feet
    • $400\pi$ cubic feet

    Correct answer: $200\pi$ cubic feet

    The volume is $200\pi$ cubic feet. The volume of a cylinder equals $\pi$ times the radius squared times the height, so $V = \pi \cdot 5^2 \cdot 8 = \pi \cdot 25 \cdot 8 = 200\pi$. Getting $80\pi$ comes from multiplying the radius by 2 instead of squaring it, since $5 \cdot 2 \cdot 8 = 80$, but the formula requires the radius to be squared.

26. A rectangular garden is 18 meters long and 11 meters wide. A walking path is built around its entire outer edge. What length of path is needed, and how does it differ from the garden's area?
    • The path (perimeter) is 58 meters, while the area is 198 square meters.
    • The path (perimeter) is 198 meters, while the area is 58 square meters.
    • The path (perimeter) is 29 meters, while the area is 198 square meters.
    • The path (perimeter) is 58 meters, while the area is 29 square meters.

    Correct answer: The path (perimeter) is 58 meters, while the area is 198 square meters.

    The path is 58 meters and the area is 198 square meters. Perimeter measures the distance around the rectangle, so it adds all four sides: $2 \times (18 + 11) = 2 \times 29 = 58$ meters. Area measures the surface enclosed, so it multiplies length by width: $18 \times 11 = 198$ square meters. Swapping the two results confuses a one-dimensional boundary length with a two-dimensional region.

27. If $\sin(\theta) = \frac{3}{5}$ and theta is an acute angle, what is the value of $\cos(\theta)$?
    • $\frac{5}{3}$
    • $\frac{3}{4}$
    • $\frac{\sqrt{2}}{2}$
    • $\frac{4}{5}$

    Correct answer: $\frac{4}{5}$

    The value is $\frac{4}{5}$. The Pythagorean identity states that $\sin^2\theta + \cos^2\theta = 1$, so $\cos^2\theta = 1 - (\frac{3}{5})^2 = 1 - \frac{9}{25} = \frac{16}{25}$, and the positive square root for an acute angle is $\frac{4}{5}$. Reporting $\frac{5}{3}$ inverts the given sine ratio rather than applying the identity, which does not produce the cosine of the same angle.

28. What is the slope of the line passing through the points (2, 3) and (6, 11)?
    • 2
    • 4
    • $\frac{1}{2}$
    • 8

    Correct answer: 2

    The slope is 2. Slope is the change in y divided by the change in x, so $\frac{11 - 3}{6 - 2}$ equals $\frac{8}{4}$, which is 2. Dividing the change in x by the change in y instead would invert the ratio and give $\frac{1}{2}$, but slope always places the vertical change over the horizontal change.

29. A line is written in slope-intercept form as $y = -3x + 7$. What is the y-intercept of this line?
    • (0, 7)
    • (0, -3)
    • (7, 0)
    • (-3, 0)

    Correct answer: (0, 7)

    The y-intercept is (0, 7). In slope-intercept form $y = mx + b$, the constant b is the y-intercept and m is the slope, so here $b = 7$ marks where the line crosses the y-axis at $x = 0$. Choosing (0, -3) confuses the slope -3 with the intercept, and writing (7, 0) places the value on the wrong axis since a y-intercept always has an x-coordinate of 0.

30. Write the equation of the line with slope 4 that passes through the point (1, 2) in slope-intercept form.
    • $y = 4x - 2$
    • $y = 4x + 2$
    • $y = 4x + 6$
    • $y = x + 4$

    Correct answer: $y = 4x - 2$

    The equation is $y = 4x - 2$. Starting from $y = mx + b$ with $m = 4$ and substituting the point (1, 2) gives $2 = 4(1) + b$, so $b = 2 - 4 = -2$. The line is therefore $y = 4x - 2$. Choosing $y = 4x + 2$ forgets to solve for b and simply reuses the point's y-value as the intercept, which does not satisfy the given point.

31. Which statement correctly describes the relationship between the lines $y = 2x + 1$ and $y = -\frac{1}{2} x + 4$?
    • They are perpendicular because their slopes are negative reciprocals.
    • They are parallel because their slopes are equal.
    • They are the same line because they share a y-intercept.
    • They never intersect because both have positive slopes.

    Correct answer: They are perpendicular because their slopes are negative reciprocals.

    The lines are perpendicular because their slopes are negative reciprocals. One slope is 2 and the other is $-\frac{1}{2}$, and multiplying them gives $2 \times -\frac{1}{2} = -1$, the defining condition for perpendicular lines. They are not parallel because parallel lines require identical slopes, and 2 does not equal $-\frac{1}{2}$.

32. What are the coordinates of the vertex of the parabola $y = (x - 3)^2 + 5$?
    • (3, 5)
    • (-3, 5)
    • (3, -5)
    • (5, 3)

    Correct answer: (3, 5)

    The vertex is (3, 5). In vertex form $y = (x - h)^2 + k$, the vertex is the point (h, k), and here $h = 3$ and $k = 5$. Note that the value subtracted inside the parentheses makes h positive, so $x - 3$ gives $h = 3$, not -3; choosing (-3, 5) misreads the sign of h inside vertex form.

33. For the parabola $y = x^2 - 6x + 8$, which way does it open and where does it cross the x-axis?
    • Opens upward, crossing the x-axis at x = 2 and x = 4
    • Opens downward, crossing the x-axis at x = 2 and x = 4
    • Opens upward, crossing the x-axis at x = -2 and x = -4
    • Opens downward, with no x-intercepts

    Correct answer: Opens upward, crossing the x-axis at x = 2 and x = 4

    The parabola opens upward and crosses the x-axis at $x = 2$ and $x = 4$. The leading coefficient on $x^2$ is positive ($a = 1$), so the parabola opens upward. Setting $y = 0$ and factoring gives $(x - 2)(x - 4) = 0$, so the x-intercepts are 2 and 4. A negative leading coefficient would open the curve downward, but none is present here.

34. A function is defined by $f(x) = 2x^2 - 5$. What is the value of $f(-3)$?
    • 13
    • -23
    • 31
    • -13

    Correct answer: 13

    The value is 13. Substituting $x = -3$ gives $f(-3) = 2(-3)^2 - 5$, and because $(-3)^2 = 9$, this becomes $2(9) - 5 = 18 - 5 = 13$. Choosing -13 squares only the 3 while wrongly keeping the negative sign inside the squaring, but squaring a negative number always produces a positive result.

35. Which set of ordered pairs represents a function?
    • {(1, 2), (3, 4), (5, 6)}
    • {(1, 2), (1, 4), (5, 6)}
    • {(2, 3), (2, 5), (2, 7)}
    • {(4, 1), (4, 2), (4, 3)}

    Correct answer: {(1, 2), (3, 4), (5, 6)}

    The set {(1, 2), (3, 4), (5, 6)} represents a function. A relation is a function only when each input (x-value) maps to exactly one output, and in this set every x-value appears just once. The set {(1, 2), (1, 4), (5, 6)} fails because the input 1 is paired with both 2 and 4, giving one input two different outputs.

36. Simplify the expression $x^5 \times x^3$ using the laws of exponents.
    • $x^8$
    • $x^{15}$
    • $x^2$
    • $2x^8$

    Correct answer: $x^8$

    The result is $x^8$. When multiplying powers that share the same base, the exponents are added, so $x^5 \times x^3$ equals $x^{5 + 3} = x^8$. Multiplying the exponents to get $x^{15}$ is the rule for raising a power to a power, not for multiplying like bases, which is why addition of exponents is correct here.

37. Rewrite $5x^{-3}$ with a positive exponent.
    • $\frac{5}{x^3}$
    • $\frac{1}{5x^3}$
    • $-5x^3$
    • $-15x$

    Correct answer: $\frac{5}{x^3}$

    The expression equals $\frac{5}{x^3}$. A negative exponent moves only its base to the denominator, so $x^{-3}$ becomes $\frac{1}{x^3}$, leaving the coefficient 5 in the numerator to give $\frac{5}{x^3}$. Writing $\frac{1}{5x^3}$ incorrectly sends the 5 to the denominator too, but the negative exponent applies only to x, not to the coefficient.

38. Add the polynomials $(3x^2 + 5x - 2)$ and $(x^2 - 4x + 7)$.
    • $4x^2 + x + 5$
    • $4x^2 + 9x + 5$
    • $2x^2 + x + 5$
    • $4x^2 - x + 9$

    Correct answer: $4x^2 + x + 5$

    The sum is $4x^2 + x + 5$. Adding polynomials means combining like terms: the $x^2$ terms $3x^2 + x^2$ give $4x^2$, the x terms $5x + (-4x)$ give x, and the constants $-2 + 7$ give 5. Combining 5x and -4x to get 9x ignores the negative sign on 4x, but those terms subtract to leave only x.

39. Multiply the binomials $(2x + 3)(x - 5)$.
    • $2x^2 - 7x - 15$
    • $2x^2 + 7x - 15$
    • $2x^2 - 13x - 15$
    • $2x^2 - 7x + 15$

    Correct answer: $2x^2 - 7x - 15$

    The product is $2x^2 - 7x - 15$. Using the distributive (FOIL) method, $2x \times x$ gives $2x^2$, $2x \times -5$ gives -10x, $3 \times x$ gives 3x, and $3 \times -5$ gives -15. Combining the middle terms $-10x + 3x$ gives -7x, so the result is $2x^2 - 7x - 15$. Getting +7x reverses the sign when combining the inner and outer products.

40. Factor the trinomial $x^2 + 9x + 20$ completely.
    • (x + 4)(x + 5)
    • (x + 2)(x + 10)
    • (x - 4)(x - 5)
    • (x + 1)(x + 20)

    Correct answer: (x + 4)(x + 5)

    The factored form is (x + 4)(x + 5). Factoring a trinomial requires two numbers that multiply to the constant 20 and add to the middle coefficient 9; those numbers are 4 and 5 because 4 times 5 is 20 and 4 plus 5 is 9. The pair 2 and 10 multiplies to 20 but adds to 12, so it does not match the required middle term.

41. Factor the difference of squares $9x^2 - 25$.
    • $(3x - 5)(3x + 5)$
    • $(3x - 5)^2$
    • $(9x - 5)(x + 5)$
    • $(3x - 25)(3x + 1)$

    Correct answer: $(3x - 5)(3x + 5)$

    The factored form is $(3x - 5)(3x + 5)$. A difference of squares $a^2 - b^2$ factors into $(a - b)(a + b)$; here $9x^2$ is $(3x)^2$ and 25 is $5^2$, so $a = 3x$ and $b = 5$, giving $(3x - 5)(3x + 5)$. Writing $(3x - 5)^2$ would expand to $9x^2 - 30x + 25$, which adds an unwanted middle term, so the two-binomial form with opposite signs is correct.

42. Solve the polynomial equation $x^2 - 5x = 0$ by factoring.
    • X = 0 and x = 5
    • X = 5 only
    • X = -5 and x = 0
    • X = 5 and x = 1

    Correct answer: X = 0 and x = 5

    The solutions are x = 0 and x = 5. Factoring out the common factor x gives x(x - 5) = 0, and the zero-product property means either x = 0 or x - 5 = 0, so x = 0 or x = 5. Dividing both sides by x to get only x = 5 discards the valid solution x = 0, which is a common error because dividing by a variable can remove a root.

43. Simplify the expression $(2x^3)^4$ using exponent rules.
    • $16x^{12}$
    • $2x^{12}$
    • $8x^7$
    • $16x^7$

    Correct answer: $16x^{12}$

    The result is $16x^{12}$. Raising a product to a power applies the exponent to each factor, so $(2x^3)^4 = 2^4 \times (x^3)^4 = 16 \times x^{3 \times 4} = 16x^{12}$. Leaving the coefficient as 2 ignores that the outer exponent must also be applied to the 2, which becomes $2^4 = 16$.

44. Simplify the rational expression $\frac{x^2 - 9}{x^2 + 7x + 12}$.
    • $\frac{x - 3}{x + 4}$
    • $\frac{x + 3}{x + 4}$
    • $\frac{x - 3}{x - 4}$
    • $\frac{x - 9}{x + 12}$

    Correct answer: $\frac{x - 3}{x + 4}$

    The simplified form is $\frac{x - 3}{x + 4}$. The numerator $x^2 - 9$ factors as $(x - 3)(x + 3)$, and the denominator $x^2 + 7x + 12$ factors as $(x + 3)(x + 4)$. The common factor $(x + 3)$ cancels, leaving $\frac{x - 3}{x + 4}$. Keeping $(x + 3)$ in the numerator instead cancels the wrong factor, since only the matching $(x + 3)$ terms cancel.

45. Multiply the rational expressions $\frac{3x}{4} \times \frac{8}{9x^2}$.
    • $\frac{2}{3x}$
    • $\frac{24x}{36x^2}$
    • $\frac{2x}{3}$
    • $\frac{3}{2x}$

    Correct answer: $\frac{2}{3x}$

    The product is $\frac{2}{3x}$. Multiplying straight across gives $\frac{3x \times 8}{4 \times 9x^2} = \frac{24x}{36x^2}$. Dividing numerator and denominator by their common factors, $\frac{24x}{36x^2}$ reduces by 12x to give $\frac{2}{3x}$. Leaving the answer as $\frac{24x}{36x^2}$ is correct in value but not simplified, while the final reduced form cancels the shared x and the numerical factor.

46. Add the rational expressions $\frac{3}{x} + \frac{5}{2x}$ and write the result as a single fraction.
    • $\frac{11}{2x}$
    • $\frac{8}{3x}$
    • $\frac{8}{2x}$
    • $\frac{15}{2x^2}$

    Correct answer: $\frac{11}{2x}$

    The sum is $\frac{11}{2x}$. The least common denominator of x and 2x is 2x, so $\frac{3}{x}$ becomes $\frac{6}{2x}$ while $\frac{5}{2x}$ stays the same. Adding the numerators gives $6 + 5 = 11$ over the common denominator, producing $\frac{11}{2x}$. Adding numerators and denominators separately to get $\frac{8}{3x}$ ignores the need for a common denominator.

47. Divide the rational expressions $\frac{x^2}{5} \div \frac{x}{10}$.
    • $2x$
    • $\frac{x}{2}$
    • $\frac{x^3}{50}$
    • $\frac{2}{x}$

    Correct answer: $2x$

    The quotient is $2x$. Dividing by a fraction means multiplying by its reciprocal, so $\frac{x^2}{5} \div \frac{x}{10}$ equals $\frac{x^2}{5} \times \frac{10}{x}$. Multiplying gives $\frac{10x^2}{5x}$, which reduces to $2x$ after canceling the common factor 5x. Multiplying the two fractions directly without flipping the second would give $\frac{x^3}{50}$, missing the reciprocal step required for division.

48. Solve the rational equation $\frac{4}{x} = \frac{8}{x + 3}$ for x.
    • X = 3
    • X = 6
    • X = -3
    • X = 12

    Correct answer: X = 3

    The solution is x = 3. Cross-multiplying the proportion gives 4(x + 3) = 8x, which expands to 4x + 12 = 8x. Subtracting 4x from both sides gives 12 = 4x, so x = 3. Choosing x = -3 would make the denominator x + 3 equal to zero, which is excluded, confirming that x = 3 is the valid answer.

49. For the rational function $f(x) = \frac{5}{x - 4}$, what is the domain restriction?
    • X cannot equal 4
    • X cannot equal -4
    • X cannot equal 0
    • X cannot equal 5

    Correct answer: X cannot equal 4

    The restriction is that x cannot equal 4. A rational function is undefined wherever its denominator equals zero, and x - 4 = 0 when x = 4, so that value must be excluded from the domain. Choosing x cannot equal -4 misreads the sign, since solving x - 4 = 0 gives positive 4, not -4.

50. Two painters working together can finish a wall in a time given by the equation $\frac{1}{x} + \frac{1}{12} = \frac{1}{4}$, where x is one painter's solo time in hours. How long would that painter take alone?
    • 6 hours
    • 8 hours
    • 3 hours
    • 16 hours

    Correct answer: 6 hours

    The painter takes 6 hours alone. Multiplying every term by the common denominator 12x clears the fractions: 12 + x = 3x. Subtracting x from both sides gives 12 = 2x, so x = 6 hours. Choosing 8 hours does not satisfy the original equation, because substituting it would make the combined rate too small to finish in 4 hours.

51. Subtract the rational expressions $\frac{7}{x + 2} - \frac{3}{x + 2}$ and simplify.
    • $\frac{4}{x + 2}$
    • $\frac{4}{2x + 4}$
    • $\frac{10}{x + 2}$
    • $\frac{4}{x^2 + 4}$

    Correct answer: $\frac{4}{x + 2}$

    The difference is $\frac{4}{x + 2}$. Because both fractions already share the denominator x + 2, only the numerators subtract: $7 - 3 = 4$, giving $\frac{4}{x + 2}$. The denominator stays the same when fractions have a common denominator, so doubling it to 2x + 4 is an error; the denominator is never added or combined during subtraction of like denominators.

52. Simplify the radical expression $\sqrt{72}$ to simplest form.
    • $6\sqrt{2}$
    • $2\sqrt{18}$
    • $36\sqrt{2}$
    • $8\sqrt{3}$

    Correct answer: $6\sqrt{2}$

    The simplified form is $6\sqrt{2}$. The largest perfect-square factor of 72 is 36, so $\sqrt{72}$ equals $\sqrt{36 \times 2}$, which separates into $\sqrt{36} \times \sqrt{2}$ = $6\sqrt{2}$. Writing $2\sqrt{18}$ is not fully simplified because 18 still contains the perfect square 9.

53. Evaluate $\sqrt[3]{64}$.
    • 4
    • 8
    • 16
    • 21.3

    Correct answer: 4

    The $\sqrt[3]{64}$ is 4. A cube root asks which number multiplied by itself three times gives the value, and 4 times 4 times 4 equals 64, so the answer is 4. Choosing 8 finds $\sqrt{64}$ instead, which uses two factors rather than the three factors a cube root requires.

54. Rewrite the expression $x^{2/3}$ using radical notation.
    • $\sqrt[3]{x^{2}}$
    • $\sqrt{x^{3}}$
    • $\sqrt[3]{x^{3}}$
    • $\sqrt{x^{2}}$

    Correct answer: $\sqrt[3]{x^{2}}$

    The expression equals $\sqrt[3]{x^{2}}$. A rational exponent of the form m/n means the numerator m is the power and the denominator n is the root index, so $x^{2/3}$ is the cube root (index 3) of x raised to the power 2. Writing $\sqrt{x^{3}}$ swaps the roles of numerator and denominator, putting the power as the index instead.

55. Add the radical expressions $5\sqrt{3}$ plus $2\sqrt{3}$.
    • $7\sqrt{3}$
    • $7\sqrt{6}$
    • $10\sqrt{3}$
    • $7\sqrt{9}$

    Correct answer: $7\sqrt{3}$

    The sum is $7\sqrt{3}$. Radical terms add only when they are like radicals sharing the same radicand, and both terms contain $\sqrt{3}$, so the coefficients add: 5 + 2 = 7, giving $7\sqrt{3}$. Adding the radicands to get $\sqrt{6}$ is incorrect, because the number under the radical stays the same when combining like radicals.

56. Multiply and simplify $\sqrt{6} \times \sqrt{10}$.
    • $2\sqrt{15}$
    • $\sqrt{60}$
    • $2\sqrt{30}$
    • $\sqrt{16}$

    Correct answer: $2\sqrt{15}$

    The product is $2\sqrt{15}$. Multiplying radicals combines the radicands under one root: $\sqrt{6} \times \sqrt{10}$ equals $\sqrt{60}$. Since 60 has the perfect-square factor 4, $\sqrt{60}$ simplifies to $2\sqrt{15}$. Leaving it as $\sqrt{60}$ is correct in value but not fully simplified.

57. Rationalize the denominator of the expression 6 divided by $\sqrt{3}$.
    • $2\sqrt{3}$
    • $6\sqrt{3}$
    • $\frac{\sqrt{3}}{2}$
    • $2\sqrt{2}$

    Correct answer: $2\sqrt{3}$

    The result is $2\sqrt{3}$. Rationalizing means multiplying numerator and denominator by $\sqrt{3}$ to clear the radical from the bottom: ($6\sqrt{3}$) divided by ($\sqrt{3} \times \sqrt{3}$) equals ($6\sqrt{3}$) divided by 3. Dividing 6 by 3 gives 2, so the answer is $2\sqrt{3}$.

58. Solve the radical equation $\sqrt{x + 5}$ = 4 for x.
    • X = 11
    • X = -1
    • X = 16
    • X = 9

    Correct answer: X = 11

    The solution is x = 11. Squaring both sides removes the radical, turning $\sqrt{x + 5}$ = 4 into x + 5 = 16. Subtracting 5 from both sides gives x = 11, and checking confirms $\sqrt{11 + 5}$ = $\sqrt{16}$ = 4. Choosing x = 16 forgets to subtract the 5, stopping at the squared value rather than solving for x.

59. Evaluate $27^{1/3}$.
    • 3
    • 9
    • 81
    • 13.5

    Correct answer: 3

    The value is 3. An exponent of 1/3 means the cube root, so $27^{1/3}$ is $\sqrt[3]{27}$, which is the number that multiplied by itself three times gives 27. Since 3 times 3 times 3 equals 27, the answer is 3. Choosing 9 mistakes the operation for squaring or dividing, but a one-third exponent specifically calls for the cube root.

60. Given $f(x) = 2x + 1$ and $g(x) = x^2$, what is the composition $f(g(3))$?
    • 19
    • 49
    • 13
    • 37

    Correct answer: 19

    The value is 19. A composition is evaluated from the inside out, so first compute $g(3) = 3^2 = 9$, then apply f to that result: $f(9) = 2(9) + 1 = 19$. Computing f(3) first and then squaring would give the wrong order of operations, since $f(g(3))$ requires g to act on 3 before f does.

61. What is the inverse function of $f(x) = 3x - 6$?
    • $f^{-1}(x) = \frac{x + 6}{3}$
    • $f^{-1}(x) = \frac{x - 6}{3}$
    • $f^{-1}(x) = 3x + 6$
    • $f^{-1}(x) = \frac{x + 3}{6}$

    Correct answer: $f^{-1}(x) = \frac{x + 6}{3}$

    The inverse is $f^{-1}(x) = \frac{x + 6}{3}$. To find an inverse, replace f(x) with y, swap x and y to get $x = 3y - 6$, then solve for y: adding 6 gives $x + 6 = 3y$, and dividing by 3 gives $y = \frac{x + 6}{3}$. Subtracting 6 instead of adding reverses the operation incorrectly, since the inverse must undo the original subtraction with addition.

62. Rewrite the logarithmic equation log base 2 of 32 = x in exponential form and find x.
    • X = 5
    • X = 16
    • X = 4
    • X = 6

    Correct answer: X = 5

    The value is x = 5. The logarithm log base 2 of 32 = x means 2 raised to the power x equals 32, and since $2^5 = 32$, the answer is x = 5. Choosing 16 confuses 32 with a different power of 2, but $2^5 = 32$ while $2^4 = 16$, so the exponent that produces 32 is 5.

63. Use the properties of logarithms to write log base b of $(xy^2)$ as a sum or difference of logarithms.
    • Log base b of x + 2 log base b of y
    • Log base b of x + log base b of $y^2$ multiplied together
    • 2 (log base b of x + log base b of y)
    • Log base b of x - 2 log base b of y

    Correct answer: Log base b of x + 2 log base b of y

    The expanded form is log base b of x + 2 log base b of y. The product rule turns the log of a product into a sum of logs, giving log base b of x + log base b of $y^2$, and the power rule then moves the exponent 2 in front of the y term, producing 2 log base b of y. Using subtraction would apply the quotient rule, which does not fit a product.

64. An investment grows according to the exponential model $A = 500(2)^t$, where t is measured in decades. What is the account value after 3 decades?
    • 4000
    • 3000
    • 1500
    • 2500

    Correct answer: 4000

    The value is 4000. Substituting $t = 3$ gives $A = 500(2)^3$, and since $2^3 = 8$, this becomes $500 \times 8 = 4000$. Multiplying 500 by 2 and then by 3 to get 3000 treats the growth as repeated multiplication by a constant rather than as an exponent, but the model raises 2 to the power $t$, as $2^t$.

65. Solve the exponential equation $5^x = 125$ for x.
    • X = 3
    • X = 25
    • X = 5
    • X = 2

    Correct answer: X = 3

    The solution is x = 3. Both sides can be written with base 5 because 125 equals $5^3$, so $5^x = 5^3$ means the exponents must be equal, giving x = 3. Choosing 25 confuses the value 125 with a square, but 5 raised to the power 3 equals 125, while 5 squared is only 25.

66. Evaluate -7 + (-12) - (-5).
    • -14
    • -24
    • -10
    • 24

    Correct answer: -14

    The result is -14. Adding two negatives, -7 + (-12), gives -19, and subtracting a negative is the same as adding its positive, so -19 - (-5) becomes -19 + 5 = -14. Treating the final subtraction as a further decrease would give -24, but subtracting a negative always moves the value upward.

67. What is the product of -6 and -9?
    • -54
    • 54
    • -15
    • 15

    Correct answer: 54

    The product is 54. Multiplying two negative numbers yields a positive result, so -6 times -9 equals positive 54. The choice -54 keeps a negative sign, but a negative times a negative cannot stay negative.

68. Evaluate -48 divided by -8.
    • 6
    • -6
    • -40
    • 40

    Correct answer: 6

    The quotient is 6. Dividing a negative by a negative produces a positive result, and 48 divided by 8 is 6, so the answer is positive 6. The choice -6 keeps the result negative, but matching signs in a quotient always give a positive value.

69. Simplify the expression |{-15}| - |{8}|.
    • -7
    • 23
    • 7
    • -23

    Correct answer: 7

    The value is 7. The absolute value of -15 is 15 and the absolute value of 8 is 8, so the expression becomes 15 - 8 = 7. The choice -7 reverses the subtraction order, but the bars are evaluated first and then 15 - 8 is computed as written.

70. What is the absolute value of the expression 4 - 11?
    • 7
    • -7
    • 15
    • -15

    Correct answer: 7

    The answer is 7. Inside the bars, 4 - 11 equals -7, and the absolute value of -7 is its distance from zero, which is 7. The choice -7 stops before applying the absolute value, but the bars require a nonnegative result.

71. Find the greatest common factor of 36 and 60.
    • 6
    • 18
    • 12
    • 180

    Correct answer: 12

    The greatest common factor is 12. The factors of 36 include 1, 2, 3, 4, 6, 9, 12, 18, and 36, while 60 has factors including 12, and 12 is the largest value dividing both evenly. The choice 6 also divides both numbers but is not the greatest such factor, since 12 is larger and still divides each.

72. What is the least common multiple of 8 and 12?
    • 96
    • 4
    • 24
    • 48

    Correct answer: 24

    The least common multiple is 24. Multiples of 8 are 8, 16, 24, 32, and multiples of 12 are 12, 24, 36, so the smallest value appearing in both lists is 24. The choice 96 is a common multiple because 8 times 12 equals 96, but it is not the least common multiple.

73. Express the fraction 18/24 in lowest terms.
    • 9/12
    • 6/8
    • 3/4
    • 2/3

    Correct answer: 3/4

    In lowest terms the fraction is 3/4. The greatest common factor of 18 and 24 is 6, and dividing both the numerator and denominator by 6 gives 18 divided by 6 equals 3 and 24 divided by 6 equals 4. The choice 9/12 divides only by 2, leaving a fraction that can still be reduced further.

74. Write the improper fraction 23/5 as a mixed number.
    • 4 3/5
    • 3 4/5
    • 5 3/4
    • 4 4/5

    Correct answer: 4 3/5

    The mixed number is 4 3/5. Dividing 23 by 5 gives a whole-number part of 4 with a remainder of 3, and the remainder becomes the new numerator over the original denominator 5. The choice 3 4/5 swaps the whole-number part and the numerator, which does not match the division result.

75. Convert the mixed number 3 2/7 to an improper fraction.
    • 6/7
    • 23/7
    • 11/7
    • 5/7

    Correct answer: 23/7

    The improper fraction is 23/7. Multiply the whole number 3 by the denominator 7 to get 21, then add the numerator 2 to reach 23, keeping the same denominator 7. The choice 6/7 multiplies the whole number by the numerator instead of the denominator, which is not the correct procedure.

76. Which symbol correctly compares the fractions 5/8 and 7/12?
    • 5/8 = 7/12
    • 5/8 < 7/12
    • 5/8 > 7/12
    • Cannot be determined

    Correct answer: 5/8 > 7/12

    The correct comparison is 5/8 > 7/12. Using a common denominator of 24, 5/8 becomes 15/24 and 7/12 becomes 14/24, and since 15/24 is larger than 14/24 the first fraction is greater. The choice 5/8 < 7/12 reverses the comparison, which the common-denominator numerators 15 and 14 contradict.

77. Convert the decimal 0.45 to a fraction in lowest terms.
    • 45/100
    • 4/5
    • 9/20
    • 45/1000

    Correct answer: 9/20

    The fraction is 9/20. The decimal 0.45 means 45 hundredths, written as 45/100, and dividing both parts by their greatest common factor 5 gives 9/20. The choice 45/100 is correct before reducing but is not in lowest terms as requested.

78. Write 0.7 as a percent.
    • 0.7%
    • 7%
    • 70%
    • 700%

    Correct answer: 70%

    The percent is 70%. Converting a decimal to a percent multiplies by 100, which moves the decimal point two places to the right, turning 0.7 into 70%. The choice 7% moves the decimal only one place, which undercounts the value by a factor of ten.

79. Write 125% as a decimal.
    • 12.5
    • 1.25
    • 0.125
    • 125

    Correct answer: 1.25

    The decimal is 1.25. Converting a percent to a decimal divides by 100, moving the decimal point two places to the left, so 125% becomes 1.25. The choice 12.5 moves the point only one place, which leaves the value ten times too large.

80. Round the number 4.768 to the nearest tenth.
    • 4.7
    • 4.8
    • 4.77
    • 5.0

    Correct answer: 4.8

    Rounded to the nearest tenth the number is 4.8. The tenths digit is 7, and the digit just after it is 6, which is 5 or greater, so the tenths digit rounds up from 7 to 8. The choice 4.7 keeps the tenths digit unchanged, ignoring that the following 6 forces a round-up.

81. Add the decimals 3.6 + 0.45 + 12.
    • 16.05
    • 4.83
    • 16.5
    • 15.81

    Correct answer: 16.05

    The sum is 16.05. Aligning the decimal points and treating 12 as 12.00 gives 3.60 + 0.45 + 12.00, and adding column by column yields 16.05. The choice 4.83 results from adding the digits without aligning place values, which mixes tenths and hundredths incorrectly.

82. Multiply 0.6 by 0.04.
    • 0.24
    • 2.4
    • 0.024
    • 0.0024

    Correct answer: 0.024

    The product is 0.024. Multiplying 6 by 4 gives 24, and since 0.6 has one decimal place and 0.04 has two, the product must have three decimal places, placing the digits as 0.024. The choice 0.24 uses only two decimal places, which misplaces the decimal point by one position.

83. What is 30% of 250?
    • 75
    • 83.3
    • 7.5
    • 750

    Correct answer: 75

    The answer is 75. Finding a percent of a number multiplies the decimal form of the percent by the number, so 0.30 times 250 equals 75. The choice 7.5 misplaces the decimal point by treating 30% as 0.03 instead of 0.30.

84. A jacket originally priced at $80 is marked down by 25%. What is the sale price?
    • $55
    • $20
    • $75
    • $60

    Correct answer: $60

    The sale price is $60. A 25% discount on $80 is 0.25 times 80, or $20, and subtracting that discount gives 80 - 20 = $60. The choice $20 reports only the amount of the discount rather than the price the shopper actually pays.

85. A town's population grew from 4,000 to 4,600 in one year. What was the percent increase?
    • 6%
    • 60%
    • 15%
    • 13%

    Correct answer: 15%

    The percent increase is 15%. The amount of increase is 4,600 - 4,000 = 600, and dividing the increase by the original amount, 600 divided by 4,000, gives 0.15, or 15%. The choice 13% incorrectly divides the increase by the new population instead of the original value.

86. The ratio of cats to dogs at a shelter is 3 to 5. If there are 24 cats, how many dogs are there?
    • 40
    • 14
    • 45
    • 8

    Correct answer: 40

    There are 40 dogs. Setting the ratio 3 cats to 5 dogs equal to 24 cats to x dogs and cross-multiplying gives 3x = 5 times 24 = 120, so x = 40. The choice 14 comes from adding the difference between 24 and 3 to 5, which ignores that the two quantities scale by the same multiplier.

87. A car travels 165 miles using 5 gallons of gas. At this rate, how many miles can it travel on 8 gallons?
    • 264
    • 33
    • 132
    • 320

    Correct answer: 264

    The car can travel 264 miles. The unit rate is 165 divided by 5, which equals 33 miles per gallon, and multiplying 33 by 8 gallons gives 264 miles. The choice 33 reports only the miles per gallon, not the total distance for 8 gallons.

88. Estimate the value of $\sqrt{50}$ to the nearest whole number.
    • 8
    • 6
    • 25
    • 7

    Correct answer: 7

    The closest whole number is 7. Since 7 squared is 49 and 8 squared is 64, $\sqrt{50}$ lies between 7 and 8, and because 50 is just above 49 it rounds to 7. The choice 25 results from dividing 50 by 2 rather than finding a number whose square is near 50.

89. Solve the linear equation $6x + 11 = 4x - 3$ for $x$.
    • -7
    • 7
    • -4
    • 4

    Correct answer: -7

    The solution is $x = -7$. Subtracting $4x$ from both sides gathers the variable terms, giving $2x + 11 = -3$, and subtracting 11 from both sides gives $2x = -14$, so $x = -7$. Subtracting $4x$ from the left while forgetting to keep the constant on the right balanced is the common slip that flips the sign and yields a positive value instead.

90. Solve the linear equation $8 - 2(3x - 1) = 4x + 30$ for $x$.
    • 2
    • -2
    • -1
    • 4

    Correct answer: -2

    The solution is $x = -2$. Distributing the -2 gives $8 - 6x + 2 = 4x + 30$, which simplifies to $10 - 6x = 4x + 30$. Adding $6x$ to both sides and subtracting 30 gives $-20 = 10x$, so $x = -2$. Distributing -2 only to the $3x$ and ignoring the -1 inside the parentheses drops a +2 term and produces a wrong value.

91. A gym charges a one-time $30 sign-up fee plus $22 per month. If a member has paid $250 in total, the equation is $30 + 22m = 250$. How many months has the member been enrolled?
    • 12
    • 11
    • 10
    • 8

    Correct answer: 10

    The member has been enrolled 10 months. Subtracting the $30 sign-up fee from both sides gives $22m = 220$, and dividing both sides by 22 gives $m = 10$. Dividing the full $250 by 22 without first removing the one-time fee overstates the count, because the sign-up fee is charged once rather than every month.

92. Solve the literal equation $P = 2L + 2W$ for $W$ in terms of $P$ and $L$.
    • $W = P - 2L$
    • $W = \frac{P - L}{2}$
    • $W = \frac{P}{2} - \frac{L}{2} - L$
    • $W = \frac{P - 2L}{2}$

    Correct answer: $W = \frac{P - 2L}{2}$

    The result is $W = \frac{P - 2L}{2}$. Subtracting $2L$ from both sides isolates the $W$ term, giving $P - 2L = 2W$, and dividing both sides by 2 gives $W = \frac{P - 2L}{2}$. Dividing only the $P$ by 2 while leaving the $2L$ unscaled fails to divide the entire side by 2, which is why every term in the numerator must be split by the same divisor.

93. Solve the proportion $\frac{x + 2}{5} = \frac{3x - 4}{9}$ for $x$.
    • $\frac{19}{3}$
    • 6
    • $\frac{19}{6}$
    • 3

    Correct answer: $\frac{19}{3}$

    The solution is $x = \frac{19}{3}$. Cross-multiplying gives $9(x + 2) = 5(3x - 4)$, which expands to $9x + 18 = 15x - 20$. Subtracting $9x$ and adding 20 to both sides gives $38 = 6x$, so $x = \frac{38}{6} = \frac{19}{3}$. Multiplying straight across the tops and bottoms instead of cross-multiplying the diagonal terms misapplies the proportion rule and gives a wrong value.

94. Solve the inequality $5x + 7 < 2x - 8$ and choose the correct solution set.
    • $x > -5$
    • $x < -5$
    • $x < 5$
    • $x > 5$

    Correct answer: $x < -5$

    The solution is $x < -5$. Subtracting $2x$ from both sides gives $3x + 7 < -8$, and subtracting 7 gives $3x < -15$, so dividing by the positive 3 gives $x < -5$ without flipping the symbol. Because the division is by a positive number, the inequality direction stays the same; reversing it would be the error reserved for dividing by a negative.

95. Solve the inequality $\frac{x}{4} - 3 \ge 1$ and choose the correct solution set.
    • $x \le 16$
    • $x \ge 8$
    • $x \ge 16$
    • $x \ge -8$

    Correct answer: $x \ge 16$

    The solution is $x \ge 16$. Adding 3 to both sides gives $\frac{x}{4} \ge 4$, and multiplying both sides by the positive 4 gives $x \ge 16$ with the symbol unchanged. Multiplying by 4 is by a positive number, so the inequality direction is preserved; only adding 3 once instead of clearing the fraction would understate the result and give 8.

96. Solve the inequality $-2(x - 4) > 6$ and choose the correct solution set.
    • $x > 1$
    • $x < -1$
    • $x > -1$
    • $x < 1$

    Correct answer: $x < 1$

    The solution is $x < 1$. Distributing the -2 gives $-2x + 8 > 6$, and subtracting 8 gives $-2x > -2$. Dividing both sides by -2 reverses the inequality symbol, producing $x < 1$. Keeping the symbol as 'greater than' after dividing by the negative 2 is the classic error, since dividing by a negative value always flips the direction.

97. A delivery van can carry at most 1,500 pounds. It already holds 480 pounds of equipment, and it will load boxes weighing 35 pounds each. The inequality is $480 + 35b \le 1500$. What is the greatest number of whole boxes it can carry?
    • 29
    • 30
    • 31
    • 44

    Correct answer: 29

    The greatest number of whole boxes is 29. Subtracting 480 from both sides gives $35b \le 1020$, and dividing by 35 gives $b \le 29.14$. Because the van can only hold whole boxes and must stay at or under the limit, the value must be rounded down to 29. Rounding up to 30 would push the load to 480 + 1050 = 1530 pounds, which exceeds the 1,500-pound maximum, so 29 is the largest count that fits.

98. Which graph on a number line correctly represents the solution to $-x + 2 \ge 5$?
    • A closed circle at -3 shading to the right
    • A closed circle at -3 shading to the left
    • An open circle at -3 shading to the left
    • An open circle at 3 shading to the right

    Correct answer: A closed circle at -3 shading to the left

    The correct picture is a closed circle at -3 shading to the left. Subtracting 2 from both sides gives $-x \ge 3$, and multiplying by -1 reverses the symbol to give $x \le -3$, so -3 is included and all smaller values qualify. A closed circle marks that -3 is part of the solution, and shading left covers values less than -3; failing to flip the symbol when multiplying by -1 would point the shading the wrong way.

99. Which ordered pair is NOT a solution to the linear inequality $y \ge 2x - 1$?
    • (0, 0)
    • (1, 3)
    • (4, 5)
    • (-2, -3)

    Correct answer: (4, 5)

    The point (4, 5) is not a solution. Substituting $x = 4$ gives $2(4) - 1 = 7$, and the inequality requires $y \ge 7$, but $y$ is only 5, so the point fails. The pair (0, 0) gives $2(0) - 1 = -1$ and $0 \ge -1$ holds, so it satisfies the inequality. A point below the boundary line $y = 2x - 1$ does not belong to the solution region for a 'greater than or equal to' inequality.

100. For the boundary of the inequality $y < 3x + 2$, how should the line be drawn and which side shaded?
     • A solid line with shading below it
     • A dashed line with shading above it
     • A solid line with shading above it
     • A dashed line with shading below it

     Correct answer: A dashed line with shading below it

     The boundary is a dashed line with shading below it. The strict 'less than' symbol means points on the line itself are not included, so the line is dashed rather than solid. Because y is less than the expression, the solution region lies below the line, where y-values are smaller. Using a solid line would wrongly include the boundary, and shading above would capture y-values that are too large.

101. Solve the absolute value equation $|5 - x| = 12$ for all values of $x$.
     • X = -7 and x = 17
     • X = 7 and x = -17
     • X = 17 only
     • X = -7 only

     Correct answer: X = -7 and x = 17

     The solutions are x = -7 and x = 17. An absolute value equal to 12 means the inside expression equals either 12 or -12: solving $5 - x = 12$ gives $x = -7$, and solving $5 - x = -12$ gives $x = 17$. Reporting only one value ignores the second branch, which is why an absolute value equation with a positive right side produces two solutions.

102. Solve the absolute value equation $2|x + 1| - 3 = 7$ for all values of $x$.
     • X = 4 and x = 6
     • X = 4 and x = -6
     • X = 5 and x = -7
     • X = -6 only

     Correct answer: X = 4 and x = -6

     The solutions are x = 4 and x = -6. First isolate the absolute value by adding 3 and dividing by 2, giving $|x + 1| = 5$. Then $x + 1 = 5$ yields $x = 4$, and $x + 1 = -5$ yields $x = -6$. Splitting into the two cases before isolating the absolute value bar, while the 2 and -3 are still attached, produces incorrect values, so the absolute value must be alone first.

103. Solve the absolute value inequality $|x - 2| < 5$ and choose the correct solution set.
     • $x < -3$ or $x > 7$
     • $-7 < x < 3$
     • $-3 < x < 7$
     • $x < 7$

     Correct answer: $-3 < x < 7$

     The solution is $-3 < x < 7$. A 'less than' absolute value inequality becomes a single compound inequality, $-5 < x - 2 < 5$, and adding 2 to all three parts gives $-3 < x < 7$. The 'less than' case always produces an 'and' interval between two bounds, while an 'or' set of two separate rays would be the result of a 'greater than' absolute value inequality instead.

104. Solve the system of equations by substitution: $y = 3x - 4$ and $2x + y = 11$.
     • X = 5, y = 1
     • X = 1, y = -1
     • X = 2, y = 7
     • X = 3, y = 5

     Correct answer: X = 3, y = 5

     The solution is x = 3 and y = 5. Substituting $3x - 4$ for $y$ in $2x + y = 11$ gives $2x + (3x - 4) = 11$, which simplifies to $5x - 4 = 11$, so $5x = 15$ and $x = 3$. Putting $x = 3$ back into $y = 3x - 4$ gives $y = 5$. The ordered pair (3, 5) satisfies both equations, while a pair that works in only one cannot solve the system.

105. How many solutions does the system $2x + 4y = 8$ and $x + 2y = 4$ have?
     • Infinitely many solutions
     • Exactly one solution
     • No solution
     • Exactly two solutions

     Correct answer: Infinitely many solutions

     The system has infinitely many solutions. Multiplying the second equation by 2 gives $2x + 4y = 8$, which is identical to the first equation, so both describe the same line. Every point on that line satisfies both equations. Concluding one unique solution assumes the lines cross at a single point, but two equations for the same line overlap completely.

106. A canoe travels 24 miles downstream in 2 hours and the same 24 miles upstream in 3 hours. With boat speed $b$ and current speed $c$, the system is $2(b + c) = 24$ and $3(b - c) = 24$. What is the speed of the boat in still water?
     • 12 mph
     • 10 mph
     • 8 mph
     • 2 mph

     Correct answer: 10 mph

     The boat's still-water speed is 10 mph. Dividing the first equation by 2 gives $b + c = 12$, and dividing the second by 3 gives $b - c = 8$. Adding these eliminates $c$, giving $2b = 20$, so $b = 10$. Subtracting instead would solve for the current speed of 2 mph, which is not what the question asks for.

107. Solve the quadratic equation $3x^2 - 27 = 0$ using the square root method.
     • X = 9 and x = -9
     • X = 3 only
     • X = 3 and x = -3
     • X = 27 and x = -27

     Correct answer: X = 3 and x = -3

     The solutions are x = 3 and x = -3. Adding 27 to both sides gives $3x^2 = 27$, and dividing by 3 gives $x^2 = 9$. Taking the square root, $\sqrt{x^2} = \sqrt{9}$, of both sides yields $x = 3$ or $x = -3$, because both values square to 9. Taking only the positive root and reporting x = 3 alone drops the negative branch that the square root method always includes.

108. Solve the quadratic equation $2x^2 + 7x + 3 = 0$ by factoring.
     • $x = \frac{1}{2}$ and $x = 3$
     • $x = -\frac{1}{2}$ and $x = 3$
     • $x = -3$ and $x = -2$
     • $x = -\frac{1}{2}$ and $x = -3$

     Correct answer: $x = -\frac{1}{2}$ and $x = -3$

     The solutions are $x = -\frac{1}{2}$ and $x = -3$. The trinomial factors into $(2x + 1)(x + 3)$ because the outer and inner products combine to $7x$ and the last terms multiply to 3. Setting $2x + 1 = 0$ gives $x = -\frac{1}{2}$, and setting $x + 3 = 0$ gives $x = -3$. Choosing the positive versions reverses the signs; the constant +3 with a middle term of +7 requires both roots to be negative.

109. Use the quadratic formula to solve $x^2 - 6x + 9 = 0$.
     • X = 3 (a single repeated root)
     • X = 3 and x = -3
     • X = 9 and x = 1
     • X = -3 (a single repeated root)

     Correct answer: X = 3 (a single repeated root)

     The solution is x = 3 as a single repeated root. With $a = 1$, $b = -6$, and $c = 9$, the discriminant $b^2 - 4ac$ equals $36 - 36 = 0$, so the formula gives $x = \frac{6 \pm 0}{2} = 3$. A discriminant of zero produces one repeated value, so listing two different roots would wrongly assume a positive discriminant that this perfect-square trinomial does not have.

110. For the quadratic equation $x^2 + 2x + 5 = 0$, what does the discriminant $b^2 - 4ac$ reveal about the solutions?
     • Two distinct real solutions, because the discriminant is positive
     • No real solutions, because the discriminant is negative
     • Exactly one real solution, because the discriminant is zero
     • Two rational solutions, because the discriminant is a perfect square

     Correct answer: No real solutions, because the discriminant is negative

     The equation has no real solutions because the discriminant is negative. With $a = 1$, $b = 2$, and $c = 5$, the discriminant $b^2 - 4ac$ equals $4 - 4(1)(5) = 4 - 20 = -16$. A negative value under the square root means no real number satisfies the equation. Concluding two real solutions would require a positive discriminant, which this equation does not have.

111. Solve the quadratic equation $x^2 + 4x - 1 = 0$ by completing the square, leaving the answer in exact radical form.
     • X = 2 plus or minus $\sqrt{5}$
     • X = -2 plus or minus $\sqrt{3}$
     • X = -2 plus or minus $\sqrt{5}$
     • X = -4 plus or minus $\sqrt{5}$

     Correct answer: X = -2 plus or minus $\sqrt{5}$

     The solutions are x = -2 plus or minus $\sqrt{5}$. Moving the constant gives $x^2 + 4x = 1$, and adding the square of half the middle coefficient, which is $(4/2)^2 = 4$, to both sides gives $x^2 + 4x + 4 = 5$. The left side becomes the perfect square $(x + 2)^2 = 5$, so $x + 2 = \pm\sqrt{5}$ and $x = -2 \pm\sqrt{5}$. Adding 2 instead of 4 to complete the square uses the wrong constant and breaks the perfect-square factoring.

112. Simplify the quotient $y^9$ divided by $y^4$ using the laws of exponents.
     • $y^5$
     • $y^{13}$
     • $y^{2.25}$
     • $y^{36}$

     Correct answer: $y^5$

     The result is $y^5$. When dividing powers that share the same base, the exponents are subtracted, so $y^9$ divided by $y^4$ equals $y^{9 - 4} = y^5$. Adding the exponents to get $y^{13}$ is the rule for multiplying like bases, not for dividing them, so subtraction is the correct operation here.

113. Simplify the expression $(m^4)^3$ using the laws of exponents.
     • $m^{12}$
     • $m^7$
     • $m^{64}$
     • $m^1$

     Correct answer: $m^{12}$

     The result is $m^{12}$. Raising a power to a power means multiplying the exponents, so $(m^4)^3$ equals $m^{4 \times 3} = m^{12}$. Adding the exponents to get $m^7$ is the rule for multiplying like bases, but a power raised to a power requires multiplication of the exponents.

114. Evaluate the expression $7^0$.
     • 1
     • 0
     • 7
     • Undefined

     Correct answer: 1

     The value is 1. Any nonzero base raised to the zero power equals 1 by the zero-exponent rule, so $7^0 = 1$. Answering 7 confuses the zero exponent with a first power, but the zero power of any nonzero number is always 1.

115. Rewrite the expression $2^{-4}$ as a positive number.
     • $\frac{1}{16}$
     • -16
     • $\frac{1}{8}$
     • -8

     Correct answer: $\frac{1}{16}$

     The value is $\frac{1}{16}$. A negative exponent means the reciprocal of the base raised to the positive exponent, so $2^{-4}$ equals $\frac{1}{2^4} = \frac{1}{16}$. Writing -16 treats the negative exponent as a negative sign on the result, but a negative exponent produces a reciprocal, not a negative value.

116. Simplify the expression $(3a^2b)(4a^3b^5)$.
     • $12a^5b^6$
     • $12a^6b^5$
     • $7a^5b^6$
     • $12a^5b^5$

     Correct answer: $12a^5b^6$

     The product is $12a^5b^6$. Multiply the coefficients 3 and 4 to get 12, then add exponents on like bases: $a^2$ times $a^3$ gives $a^5$ and $b^1$ times $b^5$ gives $b^6$. Adding the coefficients to get 7 is wrong because coefficients are multiplied, not added, when multiplying monomials.

117. Simplify the expression $(x^4 y^6)$ divided by $(x^2 y^2)$.
     • $x^2 y^4$
     • $x^2 y^3$
     • $x^6 y^8$
     • $x^8 y^{12}$

     Correct answer: $x^2 y^4$

     The result is $x^2 y^4$. Dividing like bases subtracts their exponents, so $x^4$ divided by $x^2$ gives $x^{4 - 2} = x^2$ and $y^6$ divided by $y^2$ gives $y^{6 - 2} = y^4$. Dividing $y^6$ by $y^2$ to get $y^3$ mistakenly divides the exponents, but exponents are subtracted when dividing like bases.

118. Simplify the expression $\left(\frac{2x^2}{y^3}\right)^3$.
     • $\frac{8x^6}{y^9}$
     • $\frac{6x^6}{y^9}$
     • $\frac{8x^5}{y^6}$
     • $\frac{2x^6}{y^9}$

     Correct answer: $\frac{8x^6}{y^9}$

     The result is $\frac{8x^6}{y^9}$. A power of a quotient applies the outer exponent to every factor, so the coefficient becomes $2^3 = 8$, the numerator variable becomes $x^{2 \times 3} = x^6$, and the denominator becomes $y^{3 \times 3} = y^9$. Leaving the coefficient as 6 multiplies 2 by 3 instead of cubing it, but the 2 must be raised to the third power.

119. What is the degree of the polynomial $5x^3 - 2x^4 + 7x - 1$?
     • 4
     • 3
     • 5
     • 1

     Correct answer: 4

     The degree is 4. The degree of a polynomial is the largest exponent on the variable, and here the term $-2x^4$ has the highest exponent of 4, so the polynomial has degree 4. Choosing 3 looks only at the first written term, but the degree is determined by the greatest exponent regardless of term order.

120. Subtract the polynomials $(6x^2 + 3x - 5) - (2x^2 - 4x + 1)$.
     • $4x^2 + 7x - 6$
     • $4x^2 - x - 4$
     • $4x^2 + 7x - 4$
     • $8x^2 - x - 6$

     Correct answer: $4x^2 + 7x - 6$

     The difference is $4x^2 + 7x - 6$. Subtracting a polynomial distributes the negative sign to every term, turning the second group into $-2x^2 + 4x - 1$, then combining like terms gives $6x^2 - 2x^2 = 4x^2$, $3x + 4x = 7x$, and $-5 - 1 = -6$. Getting $-x$ for the middle term fails to distribute the negative across the $-4x$, which becomes $+4x$.

121. Multiply the monomial and binomial $3x(2x^2 - 5x + 4)$.
     • $6x^3 - 15x^2 + 12x$
     • $6x^3 - 5x + 4$
     • $6x^2 - 15x^2 + 12x$
     • $5x^3 - 15x^2 + 12x$

     Correct answer: $6x^3 - 15x^2 + 12x$

     The product is $6x^3 - 15x^2 + 12x$. Distributing $3x$ to each term multiplies coefficients and adds exponents: $3x$ times $2x^2 = 6x^3$, $3x$ times $-5x = -15x^2$, and $3x$ times $4 = 12x$. Leaving the last terms unmultiplied to get $6x^3 - 5x + 4$ distributes $3x$ to only the first term, but it must reach every term inside the parentheses.

122. Expand the square of the binomial $(x + 4)^2$.
     • $x^2 + 8x + 16$
     • $x^2 + 16$
     • $x^2 + 4x + 16$
     • $x^2 + 8x + 8$

     Correct answer: $x^2 + 8x + 16$

     The expansion is $x^2 + 8x + 16$. Squaring a binomial follows $(a + b)^2 = a^2 + 2ab + b^2$, so with $a = x$ and $b = 4$ the result is $x^2 + 2(x)(4) + 16 = x^2 + 8x + 16$. Writing only $x^2 + 16$ forgets the middle term $2ab$, which a binomial square always includes.

123. Multiply the conjugate binomials $(x - 7)(x + 7)$.
     • $x^2 - 49$
     • $x^2 + 49$
     • $x^2 - 14x - 49$
     • $x^2 - 14x + 49$

     Correct answer: $x^2 - 49$

     The product is $x^2 - 49$. Multiplying conjugates of the form $(a - b)(a + b)$ gives $a^2 - b^2$ because the middle terms cancel, so here $x^2 - 49$. The choice $x^2 + 49$ uses the wrong sign, since a difference of squares produces subtraction of the second squared term, not addition.

124. Factor out the greatest common factor of $12x^3 + 18x^2$.
     • $6x^2(2x + 3)$
     • $6x(2x^2 + 3x)$
     • $2x^2(6x + 9)$
     • $6x^2(2x + 18)$

     Correct answer: $6x^2(2x + 3)$

     The factored form is $6x^2(2x + 3)$. The greatest common factor of 12 and 18 is 6, and the lowest power of $x$ shared by both terms is $x^2$, so factoring $6x^2$ leaves $2x + 3$ inside. The form $6x(2x^2 + 3x)$ pulls out only $x$ instead of $x^2$, so it is not fully factored to the greatest common factor.

125. Factor the trinomial $x^2 - 11x + 28$ completely.
     • (x - 4)(x - 7)
     • (x + 4)(x + 7)
     • (x - 2)(x - 14)
     • (x - 4)(x + 7)

     Correct answer: (x - 4)(x - 7)

     The factored form is (x - 4)(x - 7). Two numbers must multiply to 28 and add to -11, and -4 and -7 work because (-4)(-7) = 28 and -4 + (-7) = -11. The pair +4 and +7 multiplies to 28 but adds to +11, giving the wrong sign on the middle term, so both factors must be negative.

126. Factor the trinomial $2x^2 + 7x + 3$ completely.
     • (2x + 1)(x + 3)
     • (2x + 3)(x + 1)
     • (2x - 1)(x - 3)
     • (x + 1)(x + 3)

     Correct answer: (2x + 1)(x + 3)

     The factored form is $(2x + 1)(x + 3)$. Checking by expansion, $2x$ times $x = 2x^2$, the outer and inner products $6x + x = 7x$, and $1$ times $3 = 3$, matching the trinomial. The arrangement $(2x + 3)(x + 1)$ expands to $2x^2 + 5x + 3$, whose middle term is $5x$ rather than $7x$, so it does not match.

127. Factor the perfect-square trinomial $x^2 + 12x + 36$.
     • $(x + 6)^2$
     • $(x + 12)^2$
     • $(x - 6)^2$
     • $(x + 6)(x - 6)$

     Correct answer: $(x + 6)^2$

     The factored form is $(x + 6)^2$. A perfect-square trinomial $a^2 + 2ab + b^2$ factors as $(a + b)^2$; here $x^2$ is $x$ squared, 36 is 6 squared, and the middle term $12x$ equals 2 times $x$ times 6, so it factors to $(x + 6)^2$. Writing $(x + 12)^2$ would expand to $x^2 + 24x + 144$, which does not match the given trinomial.

128. Simplify the expression by combining like terms: $4x^2 + 3x - x^2 + 5x$.
     • $3x^2 + 8x$
     • $3x^2 + 2x$
     • $5x^2 + 8x$
     • $3x^4 + 8x^2$

     Correct answer: $3x^2 + 8x$

     The simplified form is $3x^2 + 8x$. Combining like terms adds the $x^2$ terms $4x^2 - x^2 = 3x^2$ and the $x$ terms $3x + 5x = 8x$. Adding $4x^2$ and $x^2$ to get $5x^2$ ignores the subtraction sign, but the second squared term is being removed, so the coefficient is 3.

129. Multiply the binomials $(3x - 2)(2x + 5)$.
     • $6x^2 + 11x - 10$
     • $6x^2 - 11x - 10$
     • $6x^2 + 11x + 10$
     • $5x^2 + 11x - 10$

     Correct answer: $6x^2 + 11x - 10$

     The product is $6x^2 + 11x - 10$. Using FOIL, $3x$ times $2x = 6x^2$, $3x$ times $5 = 15x$, $-2$ times $2x = -4x$, and $-2$ times $5 = -10$; combining $15x - 4x$ gives $11x$. Getting $-11x$ for the middle term reverses the signs of the inner and outer products, but $15x - 4x$ is positive $11x$.

130. Multiply the binomial by the trinomial $(x + 2)(x^2 + 3x + 1)$.
     • $x^3 + 5x^2 + 7x + 2$
     • $x^3 + 3x^2 + x + 2$
     • $x^3 + 5x^2 + 6x + 2$
     • $x^3 + 6x^2 + 7x + 2$

     Correct answer: $x^3 + 5x^2 + 7x + 2$

     The product is $x^3 + 5x^2 + 7x + 2$. Distributing $x$ gives $x^3 + 3x^2 + x$, distributing 2 gives $2x^2 + 6x + 2$, and combining like terms yields $x^3 + (3x^2 + 2x^2) + (x + 6x) + 2 = x^3 + 5x^2 + 7x + 2$. The choice with $3x^2$ fails to add the $2x^2$ from distributing the 2, so the squared coefficient is too small.

131. Write the polynomial $4 - 9x^2 + 2x^3 - x$ in standard form (descending powers).
     • $2x^3 - 9x^2 - x + 4$
     • $4 - x - 9x^2 + 2x^3$
     • $2x^3 - 9x^2 + x + 4$
     • $-9x^2 + 2x^3 - x + 4$

     Correct answer: $2x^3 - 9x^2 - x + 4$

     The standard form is $2x^3 - 9x^2 - x + 4$. Standard form lists terms from highest exponent to lowest, so the order is the $x^3$ term, then $x^2$, then $x$, then the constant, keeping each term's sign: $2x^3$, $-9x^2$, $-x$, $+4$. The version with $+x$ changes the sign of the $-x$ term, but rearranging must preserve each term's original sign.

132. Factor the difference of squares $x^2 - 1$ completely.
     • $(x - 1)(x + 1)$
     • $(x - 1)^2$
     • $(x + 1)^2$
     • $x(x - 1)$

     Correct answer: $(x - 1)(x + 1)$

     The factored form is $(x - 1)(x + 1)$. A difference of squares $a^2 - b^2$ factors into $(a - b)(a + b)$; here $x^2$ is x squared and 1 is 1 squared, so it becomes $(x - 1)(x + 1)$. Writing $(x - 1)^2$ would expand to $x^2 - 2x + 1$, which adds an unwanted middle term, so the conjugate pair is correct.

133. Find the value of the polynomial $P(x) = x^2 - 4x + 3$ when $x = 5$.
     • 8
     • 48
     • 23
     • -2

     Correct answer: 8

     The value is 8. Substituting $x = 5$ gives $(5)^2 - 4(5) + 3 = 25 - 20 + 3 = 8$. The answer 48 comes from squaring after multiplying or mishandling order of operations, but the exponent must be evaluated before the products are subtracted, giving 8.

134. Identify the leading coefficient of the polynomial $-3x^4 + 6x^2 - x + 9$.
     • -3
     • 9
     • 6
     • -1

     Correct answer: -3

     The leading coefficient is -3. The leading coefficient is the number multiplying the term with the highest exponent, which here is $-3x^4$, so the leading coefficient is -3. Choosing 9 picks the constant term instead, but the leading coefficient comes from the highest-degree term, not the constant.

135. Solve the polynomial equation $x^2 - 3x - 10 = 0$ by factoring.
     • $x = 5$ and $x = -2$
     • $x = -5$ and $x = 2$
     • $x = 5$ and $x = 2$
     • $x = -5$ and $x = -2$

     Correct answer: $x = 5$ and $x = -2$

     The solutions are $x = 5$ and $x = -2$. Factoring needs two numbers that multiply to -10 and add to -3, which are -5 and 2, giving $(x - 5)(x + 2) = 0$, so the zero-product property yields $x = 5$ or $x = -2$. The pair $x = -5$ and $x = 2$ comes from swapping the signs in the factors, which would give the wrong middle term.

136. Simplify the product of powers $(x^3 y^2)(x y^4) z^0$.
     • $x^4 y^6$
     • $x^4 y^6 z$
     • $x^3 y^8$
     • $x^4 y^8 z^0$

     Correct answer: $x^4 y^6$

     The result is $x^4 y^6$. Adding exponents on like bases gives $x^{3 + 1} = x^4$ and $y^{2 + 4} = y^6$, while $z^0$ equals 1 and therefore drops out entirely. Keeping z in the answer ignores that any nonzero base to the zero power is 1, so the z factor disappears.

137. Simplify the rational expression $(x^2 - 25) / (x^2 - 2x - 15)$.
     • $(x - 5) / (x - 3)$
     • $(x - 5) / (x + 3)$
     • $(x + 5) / (x - 3)$
     • $(x + 5) / (x + 3)$

     Correct answer: $(x + 5) / (x + 3)$

     The simplified form is $(x + 5) / (x + 3)$. The numerator $x^2 - 25$ factors as $(x - 5)(x + 5)$, and the denominator $x^2 - 2x - 15$ factors as $(x - 5)(x + 3)$. The common factor $(x - 5)$ cancels, leaving $(x + 5) / (x + 3)$. Choosing $(x + 5) / (x - 3)$ keeps the wrong surviving denominator factor, since the denominator factors as $(x - 5)(x + 3)$, so it is $(x + 3)$ that remains after the $(x - 5)$ terms cancel.

138. Reduce the rational expression $(6x + 18) / (x^2 - 9)$ to lowest terms.
     • $(x + 3) / 6$
     • $6 / (x + 3)$
     • $6 / (x - 3)$
     • $6 / (x^2 - 3)$

     Correct answer: $6 / (x - 3)$

     The reduced form is $6 / (x - 3)$. Factoring the numerator gives $6(x + 3)$, and factoring the denominator $x^2 - 9$ gives $(x + 3)(x - 3)$. Canceling the shared $(x + 3)$ leaves $6 / (x - 3)$. Choosing $6 / (x + 3)$ cancels the wrong factor, because it is the $(x + 3)$ pair that cancels and the $(x - 3)$ factor that remains.

139. Multiply the rational expressions $(x + 2) / (x - 5)$ times $(x^2 - 25) / (x^2 - 4)$.
     • $(x - 5) / (x + 2)$
     • $(x + 5) / (x - 2)$
     • $(x + 5) / (x + 2)$
     • $(x - 2) / (x + 5)$

     Correct answer: $(x + 5) / (x - 2)$

     The product is $(x + 5) / (x - 2)$. Factoring gives $(x + 2) / (x - 5)$ times $(x - 5)(x + 5) / ((x - 2)(x + 2))$. The $(x - 5)$ factors cancel and the $(x + 2)$ factors cancel, leaving $(x + 5) / (x - 2)$. Choosing $(x - 5) / (x + 2)$ fails to cancel either matched pair, leaving factors that should have been removed.

140. Multiply the rational expressions $(10 / (x + 4))$ times $((x^2 + 4x) / 15)$.
     • $2 / (3x)$
     • $3x / 2$
     • $10x / 15$
     • $2x / 3$

     Correct answer: $2x / 3$

     The product is $2x / 3$. Factoring $x^2 + 4x$ as $x(x + 4)$ gives $(10 / (x + 4))$ times $(x(x + 4) / 15)$. The $(x + 4)$ factors cancel, leaving $10x / 15$, which reduces to $2x / 3$ after dividing top and bottom by 5. Leaving $10x / 15$ is correct in value but not in lowest terms as required.

141. Divide the rational expressions $((x^2 - 1) / (x + 3))$ divided by $((x - 1) / (x + 3))$.
     • $(x^2 - 1) / (x - 1)$
     • $x - 1$
     • $(x + 1) / (x + 3)$
     • $x + 1$

     Correct answer: $x + 1$

     The quotient is $x + 1$. Dividing means multiplying by the reciprocal: $((x^2 - 1) / (x + 3))$ times $((x + 3) / (x - 1))$. The $(x + 3)$ factors cancel, and factoring $x^2 - 1$ as $(x - 1)(x + 1)$ lets the $(x - 1)$ factors cancel, leaving $x + 1$. Choosing $(x + 1) / (x + 3)$ forgets to cancel the $(x + 3)$ introduced by the reciprocal.

142. Divide the rational expressions $(8 / (x - 6))$ divided by $(12 / (x^2 - 36))$.
     • $3(x + 6) / 2$
     • $2(x + 6) / 3$
     • $2(x - 6) / 3$
     • $8(x + 6) / 12$

     Correct answer: $2(x + 6) / 3$

     The quotient is $2(x + 6) / 3$. Multiplying by the reciprocal gives $(8 / (x - 6))$ times $((x^2 - 36) / 12)$. Factoring $x^2 - 36$ as $(x - 6)(x + 6)$ cancels the $(x - 6)$, leaving $8(x + 6) / 12$, which reduces to $2(x + 6) / 3$ by dividing the numbers by 4. Choosing $8(x + 6) / 12$ stops before reducing the numerical factor.

143. Add the rational expressions $5/(x + 1) + 2/(x - 1)$ and write the result as a single fraction.
     • $(7x - 3) / (x^2 - 1)$
     • $7 / (x^2 - 1)$
     • $(7x + 3) / (x^2 - 1)$
     • $(7x - 3) / (2x)$

     Correct answer: $(7x - 3) / (x^2 - 1)$

     The sum is $(7x - 3) / (x^2 - 1)$. The least common denominator is $(x + 1)(x - 1)$, so the numerators become $5(x - 1) + 2(x + 1) = 5x - 5 + 2x + 2 = 7x - 3$ over $(x + 1)(x - 1)$, which is $x^2 - 1$. Choosing $7 / (x^2 - 1)$ wrongly adds the numerators 5 and 2 without first multiplying by the missing factors.

144. Add the rational expressions $4/(3x) + 1/(2x)$ and write the result as a single fraction.
     • $5 / (5x)$
     • $11 / (6x)$
     • $5 / (6x)$
     • $11 / (5x)$

     Correct answer: $11 / (6x)$

     The sum is $11 / (6x)$. The least common denominator of $3x$ and $2x$ is $6x$, so $4/(3x)$ becomes $8/(6x)$ and $1/(2x)$ becomes $3/(6x)$. Adding the numerators gives $8 + 3 = 11$ over $6x$, producing $11 / (6x)$. Choosing $5 / (5x)$ adds the numerators and denominators directly, ignoring the need for a common denominator.

145. Subtract the rational expressions $3/(x - 2) - 1/(x + 2)$ and write the result as a single fraction.
     • $2 / (x^2 - 4)$
     • $(2x - 8) / (x^2 - 4)$
     • $(2x + 8) / (x^2 - 4)$
     • $(2x + 8) / (2x)$

     Correct answer: $(2x + 8) / (x^2 - 4)$

     The difference is $(2x + 8) / (x^2 - 4)$. Using the common denominator $(x - 2)(x + 2)$, the numerators become $3(x + 2) - 1(x - 2) = 3x + 6 - x + 2 = 2x + 8$ over $(x - 2)(x + 2)$, which equals $x^2 - 4$. Choosing $(2x - 8) / (x^2 - 4)$ mishandles the distribution of the minus sign across the second numerator.

146. Subtract the rational expressions $(x + 6)/(x + 3) - 3/(x + 3)$ and simplify.
     • $(x + 9) / (x + 3)$
     • $(x + 3) / (x + 3)$
     • $x / (x + 3)$
     • 1

     Correct answer: 1

     The difference is 1. Because both fractions share the denominator $x + 3$, only the numerators subtract: $(x + 6) - 3 = x + 3$, giving $(x + 3) / (x + 3)$. Since the numerator now matches the denominator, the expression equals 1. Choosing $x / (x + 3)$ subtracts only the constants and overlooks that the entire $x + 3$ numerator cancels with the denominator.

147. Solve the rational equation $6/x = 9/(x + 2)$ for $x$.
     • $x = -4$
     • $x = 4$
     • $x = 2$
     • $x = 6$

     Correct answer: $x = 4$

     The solution is $x = 4$. Cross-multiplying the proportion gives $6(x + 2) = 9x$, which expands to $6x + 12 = 9x$. Subtracting $6x$ from both sides gives $12 = 3x$, so $x = 4$. Choosing $x = 2$ does not satisfy the equation, since substituting it makes the two sides unequal.

148. Solve the rational equation $1/x + 1/4 = 1/2$ for $x$.
     • $x = 6$
     • $x = 2$
     • $x = 4$
     • $x = 8$

     Correct answer: $x = 4$

     The solution is $x = 4$. Multiplying every term by the common denominator $4x$ clears the fractions: $4 + x = 2x$. Subtracting $x$ from both sides gives $4 = x$, so $x = 4$. Choosing $x = 2$ would make the left side $1/2 + 1/4$, which is too large, so it fails the original equation.

149. Solve the rational equation $5/(x - 3) = 10$ for $x$.
     • $x = 3.5$
     • $x = 2.5$
     • $x = 4$
     • $x = 8$

     Correct answer: $x = 3.5$

     The solution is $x = 3.5$. Multiplying both sides by $(x - 3)$ gives $5 = 10(x - 3)$, which expands to $5 = 10x - 30$. Adding 30 to both sides gives $35 = 10x$, so $x = 3.5$. Choosing $x = 2.5$ reverses the sign when solving, since $35$ divided by $10$ is positive 3.5, not 2.5.

150. Solve the rational equation $(x + 4)/(x - 1) = 2$ for $x$.
     • $x = 3$
     • $x = -6$
     • $x = 6$
     • $x = 2$

     Correct answer: $x = 6$

     The solution is $x = 6$. Multiplying both sides by $(x - 1)$ gives $x + 4 = 2(x - 1)$, which expands to $x + 4 = 2x - 2$. Subtracting $x$ and adding 2 to both sides gives $6 = x$, so $x = 6$. Choosing $x = 3$ does not satisfy the equation, since $(3 + 4)/(3 - 1)$ equals 3.5, not 2.

151. Solve the rational equation $2/(x - 1) + 3/(x - 1) = 5$ for $x$.
     • $x = 0$
     • $x = 1$
     • $x = 6$
     • $x = 2$

     Correct answer: $x = 2$

     The solution is $x = 2$. Since both fractions share the denominator $x - 1$, the left side combines to $5/(x - 1) = 5$. Multiplying both sides by $(x - 1)$ gives $5 = 5(x - 1)$, so $1 = x - 1$ and $x = 2$. Choosing $x = 1$ is rejected because it makes the denominator zero, which is excluded.

152. Solve the rational equation $3/(x + 2) = 1/(x - 2)$ for $x$.
     • $x = 2$
     • $x = -4$
     • $x = 4$
     • $x = 8$

     Correct answer: $x = 4$

     The solution is $x = 4$. Cross-multiplying gives $3(x - 2) = 1(x + 2)$, which expands to $3x - 6 = x + 2$. Subtracting $x$ and adding 6 to both sides gives $2x = 8$, so $x = 4$. Choosing $x = 2$ is rejected because it makes the denominator $x - 2$ equal to zero, which is excluded.

153. Identify the value that must be excluded from the solution of any rational equation containing the term $7/(x - 9)$.
     • $x = 9$
     • $x = -9$
     • $x = 7$
     • $x = 0$

     Correct answer: $x = 9$

     The excluded value is $x = 9$. A rational expression is undefined wherever its denominator equals zero, and $x - 9 = 0$ when $x = 9$, so that value can never be a solution. Choosing $x = -9$ misreads the sign, since solving $x - 9 = 0$ gives positive 9, not negative 9.

154. For the rational function $g(x) = (x + 5) / (x^2 - 16)$, which values are excluded from the domain?
     • $x = 4$ only
     • $x = -5$ only
     • $x = 16$ only
     • $x = 4$ and $x = -4$

     Correct answer: $x = 4$ and $x = -4$

     The excluded values are $x = 4$ and $x = -4$. The domain excludes any input making the denominator zero, and $x^2 - 16 = 0$ factors as $(x - 4)(x + 4) = 0$, giving $x = 4$ and $x = -4$. Choosing $x = -5$ only confuses the numerator's zero with a domain restriction, but only the denominator determines excluded values.

155. A boat travels 30 miles downstream in the same time it takes to travel 18 miles upstream. The current flows at 4 miles per hour. Using the equation $30/(b + 4) = 18/(b - 4)$, find the boat's speed $b$ in still water.
     • 16 mph
     • 12 mph
     • 8 mph
     • 24 mph

     Correct answer: 16 mph

     The boat's still-water speed is 16 mph. Cross-multiplying gives $30(b - 4) = 18(b + 4)$, which expands to $30b - 120 = 18b + 72$. Subtracting $18b$ and adding 120 gives $12b = 192$, so $b = 16$ mph. Choosing 12 mph does not satisfy the equation, since the downstream and upstream times would no longer be equal.

156. One pipe can fill a tank in 6 hours and a second pipe can fill it in 3 hours. Using $1/6 + 1/3 = 1/t$, how long does it take both pipes working together?
     • 2 hours
     • 1 hour
     • 4.5 hours
     • 9 hours

     Correct answer: 2 hours

     Together the pipes take 2 hours. Adding the rates $1/6 + 1/3$ with common denominator 6 gives $1/6 + 2/6 = 3/6$, which simplifies to $1/2$, so $1/t = 1/2$ and $t = 2$ hours. Choosing 9 hours adds the individual times rather than the rates, which is not how combined work is computed.

157. Simplify the complex fraction $(1/2 + 1/4)$ divided by $(3/4)$.
     • $3/4$
     • 1
     • $1/2$
     • 2

     Correct answer: 1

     The simplified value is 1. The numerator $1/2 + 1/4$ combines to $3/4$ using a common denominator of 4, so the expression becomes $(3/4)$ divided by $(3/4)$. Dividing a quantity by itself gives 1. Choosing $3/4$ reports only the combined numerator and forgets to carry out the division by $3/4$.

158. Simplify the complex fraction $(x / (x + 1))$ divided by $(x^2 / (x + 1))$.
     • $x$
     • $1 / x$
     • $x / (x + 1)$
     • $x^2$

     Correct answer: $1 / x$

     The simplified form is $1 / x$. Rewriting the division as multiplication by the reciprocal gives $(x / (x + 1))$ times $((x + 1) / x^2)$. The $(x + 1)$ factors cancel and one $x$ cancels from $x$ and $x^2$, leaving $1 / x$. Choosing $x$ inverts the reduction, since the larger power $x^2$ sits in the denominator after flipping.

159. Multiply the rational expressions $((x^2 + 6x + 9) / (x + 1))$ times $((x + 1) / (x + 3))$.
     • $x + 3$
     • $x + 1$
     • $(x + 3) / (x + 1)$
     • $(x + 3)^2$

     Correct answer: $x + 3$

     The product is $x + 3$. Factoring $x^2 + 6x + 9$ as $(x + 3)(x + 3)$ gives $((x + 3)(x + 3) / (x + 1))$ times $((x + 1) / (x + 3))$. The $(x + 1)$ factors cancel and one $(x + 3)$ cancels, leaving $x + 3$. Choosing $(x + 3)^2$ fails to cancel the second $(x + 3)$ factor against the one in the second denominator.

160. Solve the rational equation $9/(x^2 - 9) = 1/(x - 3)$ for $x$.
     • $x = 6$
     • $x = 3$
     • $x = -3$
     • $x = 0$

     Correct answer: $x = 6$

     The solution is $x = 6$. Factoring $x^2 - 9$ as $(x - 3)(x + 3)$ and multiplying both sides by it gives $9 = 1(x + 3)$, so $x + 3 = 9$ and $x = 6$. Choosing $x = 3$ is rejected because it makes the denominator zero, but $x = 6$ keeps every denominator nonzero and satisfies the original equation.

161. Add the rational expressions $2/x + 3$ and write the result as a single fraction.
     • $(2x + 3) / x$
     • $(2 + 3) / x$
     • $5 / x$
     • $(3x + 2) / x$

     Correct answer: $(3x + 2) / x$

     The sum is $(3x + 2) / x$. Writing 3 as $3x/x$ gives a common denominator of $x$, so $2/x + 3x/x$ combines to $(2 + 3x) / x$, which is the same as $(3x + 2) / x$. Choosing $5 / x$ incorrectly treats 3 as $3/x$ and adds only the numerators, ignoring that the whole number 3 equals $3x$ over $x$.

162. Simplify the radical expression $\sqrt{50}$ to simplest form.
     • $5\sqrt{2}$
     • $2\sqrt{5}$
     • $25\sqrt{2}$
     • $10\sqrt{5}$

     Correct answer: $5\sqrt{2}$

     The simplified form is $5\sqrt{2}$. The largest perfect-square factor of 50 is 25, so $\sqrt{50}$ equals $\sqrt{25 \times 2}$, which separates into $\sqrt{25} \times \sqrt{2}$ = $5\sqrt{2}$. Writing $2\sqrt{5}$ mistakenly pulls out the 2 instead of the perfect square 25.

163. Evaluate $\sqrt[3]{125}$.
     • 5
     • 25
     • 15
     • 11.2

     Correct answer: 5

     The $\sqrt[3]{125}$ is 5. A cube root asks which number multiplied by itself three times gives the value, and 5 times 5 times 5 equals 125, so the answer is 5. Choosing 25 finds $\sqrt{625}$ or mistakenly squares, but a cube root requires three equal factors, not two.

164. Simplify the quotient $\sqrt{48}$ divided by $\sqrt{3}$.
     • 4
     • 16
     • $\sqrt{16}$
     • $2\sqrt{3}$

     Correct answer: 4

     The quotient equals 4. Dividing radicals combines the radicands under one root, so $\sqrt{48}$ divided by $\sqrt{3}$ equals $\sqrt{48 / 3}$ = $\sqrt{16}$, which is 4. Leaving it as $\sqrt{16}$ is correct in value but not fully evaluated, since 16 is a perfect square.

165. Subtract the radical expressions $9\sqrt{5}$ minus $4\sqrt{5}$.
     • $5\sqrt{5}$
     • $5\sqrt{0}$
     • $13\sqrt{5}$
     • $5\sqrt{10}$

     Correct answer: $5\sqrt{5}$

     The difference is $5\sqrt{5}$. Like radicals share the same radicand, and both terms contain $\sqrt{5}$, so the coefficients subtract: 9 - 4 = 5, giving $5\sqrt{5}$. The radicand stays $\sqrt{5}$ because only the coefficients change when combining like radicals.

166. Simplify the radical expression $\sqrt{12}$ plus $\sqrt{27}$.
     • $5\sqrt{3}$
     • $\sqrt{39}$
     • $6\sqrt{3}$
     • $5\sqrt{6}$

     Correct answer: $5\sqrt{3}$

     The sum is $5\sqrt{3}$. First simplify each term: $\sqrt{12}$ = $2\sqrt{3}$, and $\sqrt{27}$ = $3\sqrt{3}$. Now they are like radicals, so adding the coefficients 2 + 3 gives $5\sqrt{3}$. Adding the radicands to get $\sqrt{39}$ ignores that the terms must first be simplified to combine.

167. Rationalize the denominator of the expression 10 divided by $\sqrt{5}$.
     • $2\sqrt{5}$
     • $10\sqrt{5}$
     • $\frac{\sqrt{5}}{2}$
     • $2\sqrt{2}$

     Correct answer: $2\sqrt{5}$

     The result is $2\sqrt{5}$. Rationalizing means multiplying numerator and denominator by $\sqrt{5}$: ($10\sqrt{5}$) divided by ($\sqrt{5} \times \sqrt{5}$) equals ($10\sqrt{5}$) divided by 5. Dividing 10 by 5 gives 2, so the answer is $2\sqrt{5}$.

168. Solve the radical equation $\sqrt{2x - 1}$ = 5 for x.
     • X = 13
     • X = 12
     • X = 26
     • X = 3

     Correct answer: X = 13

     The solution is $x = 13$. Squaring both sides turns $\sqrt{2x - 1}$ = 5 into $2x - 1 = 25$. Adding 1 gives $2x = 26$, and dividing by 2 gives $x = 13$; checking confirms $\sqrt{2 \times 13 - 1}$ = $\sqrt{25}$ = 5. Choosing $x = 26$ stops after isolating $2x$ and forgets to divide by 2.

169. Evaluate $16^{3/4}$.
     • 8
     • 12
     • 64
     • 2

     Correct answer: 8

     The value is 8. A rational exponent $m/n$ means take the n-th root and raise to the m power, so $16^{3/4}$ is $\sqrt[4]{16}$, raised to the third power. The $\sqrt[4]{16}$ is 2 because 2 times 2 times 2 times 2 = 16, and 2 cubed is 8. Choosing 64 mistakenly cubes 16 first without taking the fourth root.

170. Simplify the radical expression $\sqrt{98}$.
     • $7\sqrt{2}$
     • $2\sqrt{49}$
     • $49\sqrt{2}$
     • $7\sqrt{7}$

     Correct answer: $7\sqrt{2}$

     The simplified form is $7\sqrt{2}$. The largest perfect-square factor of 98 is 49, so $\sqrt{98}$ equals $\sqrt{49 \times 2}$ = $\sqrt{49} \times \sqrt{2}$ = $7\sqrt{2}$. Writing $2\sqrt{49}$ leaves a perfect square under the radical, so it is not fully simplified.

171. Evaluate $\sqrt[4]{81}$.
     • 3
     • 9
     • 27
     • 20.25

     Correct answer: 3

     The $\sqrt[4]{81}$ is 3. A fourth root asks which number multiplied by itself four times gives the value, and 3 times 3 times 3 times 3 equals 81, so the answer is 3. Choosing 9 finds $\sqrt{81}$ instead, which uses only two equal factors rather than the four a fourth root requires.

172. Multiply and simplify $2\sqrt{3}$, times $5\sqrt{6}$.
     • $30\sqrt{2}$
     • $10\sqrt{18}$
     • $7\sqrt{9}$
     • $10\sqrt{2}$

     Correct answer: $30\sqrt{2}$

     The product is $30\sqrt{2}$. Multiply the coefficients (2 times 5 = 10) and the radicands ($\sqrt{3} \times \sqrt{6}$ = $\sqrt{18}$), giving $10\sqrt{18}$. Since 18 has the perfect-square factor 9, $\sqrt{18}$ = $3\sqrt{2}$, so 10 times 3 = 30, giving $30\sqrt{2}$.

173. Rewrite the radical expression $\sqrt[5]{x^{3}}$ using a rational exponent.
     • $x^{3/5}$
     • $x^{5/3}$
     • $x^{3+5}$
     • $x^{15}$

     Correct answer: $x^{3/5}$

     The expression equals $x^{3/5}$. A radical converts to a rational exponent where the power inside becomes the numerator and the root index becomes the denominator, so the fifth root (index 5) of x raised to the 3 power is $x^{3/5}$. Writing $x^{5/3}$ swaps the index and the power, putting the root in the wrong position.

174. Simplify $\sqrt[3]{54}$ to simplest form.
     • $3\sqrt[3]{2}$
     • $2\sqrt[3]{3}$
     • $9\sqrt[3]{6}$
     • $3\sqrt[3]{6}$

     Correct answer: $3\sqrt[3]{2}$

     The simplified form is $3\sqrt[3]{2}$. The largest perfect-cube factor of 54 is 27, so $\sqrt[3]{54}$ equals $\sqrt[3]{27 \times 2}$ = $\sqrt[3]{27} \times \sqrt[3]{2}$ = $3\sqrt[3]{2}$. With cube roots you must look for a perfect cube factor like 27, not a perfect square.

175. Rationalize the denominator of the expression 4 divided by ($\sqrt{7}$ minus $\sqrt{3}$).
     • $\sqrt{7}$ plus $\sqrt{3}$
     • $\sqrt{7}$ minus $\sqrt{3}$
     • 4 times ($\sqrt{7}$ plus $\sqrt{3}$)
     • $\sqrt{7}$ plus $\sqrt{3}$, all divided by 4

     Correct answer: $\sqrt{7}$ plus $\sqrt{3}$

     The result is $\sqrt{7}$ plus $\sqrt{3}$. Multiply numerator and denominator by the conjugate $\sqrt{7}$ plus $\sqrt{3}$, which makes the denominator $(\sqrt{7})^2 - (\sqrt{3})^2 = 7 - 3 = 4$. The numerator becomes 4 times ($\sqrt{7}$ plus $\sqrt{3}$), and dividing by 4 leaves $\sqrt{7}$ plus $\sqrt{3}$.

176. Simplify the radical expression $\sqrt{75}$ minus $\sqrt{12}$.
     • $3\sqrt{3}$
     • $\sqrt{63}$
     • $7\sqrt{3}$
     • $2\sqrt{3}$

     Correct answer: $3\sqrt{3}$

     The difference is $3\sqrt{3}$. Simplify each term first: $\sqrt{75}$ = $5\sqrt{3}$ and $\sqrt{12}$ = $2\sqrt{3}$. As like radicals, subtract the coefficients 5 - 2 = 3, giving $3\sqrt{3}$. Subtracting the radicands to get $\sqrt{63}$ skips the required simplification step.

177. Evaluate the expression $8^{2/3}$.
     • 4
     • 16
     • 6
     • 2.67

     Correct answer: 4

     The value is 4. The exponent $2/3$ means take the cube root and then raise to the second power, so $8^{2/3}$ is $\sqrt[3]{8}$ (which is 2) squared. Since 2 squared is 4, the answer is 4. Choosing 16 squares 8 first without taking the cube root, which ignores the denominator of the exponent.

178. Simplify the expression $\sqrt{8} \times \sqrt{2}$.
     • 4
     • $\sqrt{16}$
     • $2\sqrt{8}$
     • 16

     Correct answer: 4

     The product is 4. Multiplying radicals combines the radicands under one root, so $\sqrt{8} \times \sqrt{2}$ equals $\sqrt{16}$, and $\sqrt{16}$ is 4. Leaving the answer as $\sqrt{16}$ is correct in value but not fully evaluated since 16 is a perfect square.

179. Solve the radical equation $\sqrt[3]{x - 2}$ = 3 for x.
     • X = 29
     • X = 11
     • X = 5
     • X = 27

     Correct answer: X = 29

     The solution is x = 29. To undo a cube root, cube both sides, turning $\sqrt[3]{x - 2}$ = 3 into x - 2 = 27. Adding 2 gives x = 29; checking confirms $\sqrt[3]{29 - 2}$ = $\sqrt[3]{27}$ = 3. Choosing x = 11 mistakenly squares the 3 instead of cubing it, since the equation involves a cube root.

180. Simplify the radical expression $\sqrt{45}$ divided by $\sqrt{5}$.
     • 3
     • 9
     • $\sqrt{9}$
     • $3\sqrt{5}$

     Correct answer: 3

     The quotient is 3. Dividing radicals combines the radicands, so $\sqrt{45}$ divided by $\sqrt{5}$ equals $\sqrt{45 / 5}$ = $\sqrt{9}$, which is 3. Leaving it as $\sqrt{9}$ is correct in value but not fully evaluated since 9 is a perfect square.

181. Simplify the expression $\sqrt{x^{6}}$, assuming x is positive.
     • $x^3$
     • $x^2$
     • $x^{36}$
     • $\sqrt{x}$ times 6

     Correct answer: $x^3$

     The expression equals $x^3$. Taking a square root divides the exponent by 2, so $\sqrt{x^{6}}$ is $x^{6/2} = x^3$. Choosing $x^2$ mistakenly subtracts 2 from the exponent or divides by 3, but a square root corresponds to dividing the exponent by the index 2.

182. Simplify the radical expression $3\sqrt{18}$.
     • $9\sqrt{2}$
     • $3\sqrt{9}$
     • $9\sqrt{9}$
     • $6\sqrt{3}$

     Correct answer: $9\sqrt{2}$

     The simplified form is $9\sqrt{2}$. The $\sqrt{18}$ simplifies because 18 = 9 times 2, giving $3\sqrt{2}$; multiplying by the coefficient 3 outside gives 3 times 3 = 9, so the result is $9\sqrt{2}$. Forgetting to multiply the outside 3 by the new coefficient leaves the answer unfinished.

183. Rationalize the denominator of the expression $\sqrt{2}$ divided by $\sqrt{5}$.
     • $\frac{\sqrt{10}}{5}$
     • $\frac{\sqrt{10}}{2}$
     • $\frac{\sqrt{7}}{5}$
     • 2 over $\sqrt{5}$

     Correct answer: $\frac{\sqrt{10}}{5}$

     The result is $\frac{\sqrt{10}}{5}$. Multiply numerator and denominator by $\sqrt{5}$: the numerator becomes $\sqrt{2} \times \sqrt{5}$ = $\sqrt{10}$, and the denominator becomes $\sqrt{5} \times \sqrt{5}$ = 5. So the rationalized form is $\frac{\sqrt{10}}{5}$, which clears the radical from the denominator.

184. Evaluate $\sqrt[3]{-8}$.
     • -2
     • 2
     • -4
     • Not a real number

     Correct answer: -2

     The $\sqrt[3]{-8}$ is -2. Odd roots of negative numbers are real, and the number that multiplied by itself three times gives negative 8 is -2, since -2 times -2 times -2 = -8. The answer is not 2 because a positive cube would give positive 8, and odd roots preserve the negative sign.

185. Simplify the radical expression $\sqrt[3]{16}$ to simplest form.
     • $2\sqrt[3]{2}$
     • $4\sqrt[3]{2}$
     • $2\sqrt[3]{4}$
     • $8\sqrt[3]{2}$

     Correct answer: $2\sqrt[3]{2}$

     The simplified form is $2\sqrt[3]{2}$. The largest perfect-cube factor of 16 is 8, so $\sqrt[3]{16}$ equals $\sqrt[3]{8 \times 2}$ = $\sqrt[3]{8} \times \sqrt[3]{2}$ = $2\sqrt[3]{2}$. Looking for a perfect-square factor instead would not simplify a cube root correctly.

186. Solve the radical equation $\sqrt{3x + 4}$ = $\sqrt{x + 10}$ for x.
     • X = 3
     • X = 7
     • X = 14
     • X = 6

     Correct answer: X = 3

     The solution is x = 3. Squaring both sides removes the radicals because the two square roots are equal, giving 3x + 4 = x + 10. Subtracting x from both sides gives 2x + 4 = 10, then subtracting 4 gives 2x = 6 and x = 3; checking confirms both sides equal $\sqrt{13}$. Choosing x = 7 misadds the constants when solving.

187. Evaluate log base 3 of 81.
     • 27
     • 3
     • 9
     • 4

     Correct answer: 4

     The value is 4. The expression log base 3 of 81 asks for the exponent that turns 3 into 81, and since 3 raised to the power 4 equals 81, the answer is 4. Choosing 27 confuses 81 with 3 cubed, but $3^3$ equals 27 while $3^4$ equals 81.

188. Given f(x) = x - 4 and g(x) = 2x, what is the composition g(f(7))?
     • 18
     • 10
     • 6
     • 3

     Correct answer: 6

     The value is 6. Composition works from the inside out, so first evaluate f(7) = 7 - 4 = 3, then apply g to that result: g(3) = 2(3) = 6. Doubling 7 first and then subtracting 4 to get 10 reverses the required order, since g(f(7)) needs f to act on 7 before g does.

189. Rewrite the exponential statement $4^3 = 64$ in logarithmic form.
     • Log base 3 of 64 = 4
     • Log base 64 of 4 = 3
     • Log base 4 of 64 = 3
     • Log base 4 of 3 = 64

     Correct answer: Log base 4 of 64 = 3

     The correct form is log base 4 of 64 = 3. An exponential statement $\text{base}^{\text{exponent}} = \text{result}$ converts to a logarithm where the base stays the base, the result becomes the argument, and the exponent becomes the value, giving log base 4 of 64 = 3. Writing log base 3 of 64 = 4 swaps the exponent and base incorrectly.

190. A bacteria population is modeled by $P = 200(3)^t$, where t is in hours. What is the population after 2 hours?
     • 1200
     • 1800
     • 600
     • 400

     Correct answer: 1800

     The value is 1800. Substituting t = 2 gives $P = 200(3)^2$, and since $3^2 = 9$, this becomes 200 times 9 = 1800. Multiplying 200 by 3 and then by 2 to get 1200 treats the growth as a constant multiplier rather than an exponent, but the model raises 3 to the power t.

191. Use the quotient rule for logarithms to expand log base 5 of (x / 7).
     • Log base 5 of x + log base 5 of 7
     • Log base 5 of x divided by log base 5 of 7
     • Log base 5 of 7 - log base 5 of x
     • Log base 5 of x - log base 5 of 7

     Correct answer: Log base 5 of x - log base 5 of 7

     The expanded form is log base 5 of x - log base 5 of 7. The quotient rule states that the log of a quotient equals the log of the numerator minus the log of the denominator, so log base 5 of (x / 7) becomes log base 5 of x - log base 5 of 7. Using addition would apply the product rule, which does not fit a quotient.

192. Solve the logarithmic equation log base 4 of x = 3 for x.
     • 12
     • 81
     • 64
     • 7

     Correct answer: 64

     The solution is x = 64. The equation log base 4 of x = 3 converts to exponential form as 4 raised to the power 3 equals x, and since $4^3 = 64$, the value of x is 64. Multiplying 4 by 3 to get 12 treats the relationship as multiplication, but the logarithm makes 3 an exponent on the base 4.

193. What is the inverse function of f(x) = (x / 2) + 5?
     • F inverse (x) = 2(x - 5)
     • F inverse (x) = 2x - 5
     • F inverse (x) = (x - 5) / 2
     • F inverse (x) = 2x + 5

     Correct answer: F inverse (x) = 2(x - 5)

     The inverse is f inverse (x) = 2(x - 5). To find an inverse, write y = (x / 2) + 5, swap x and y to get x = (y / 2) + 5, subtract 5 to get x - 5 = y / 2, then multiply by 2 to get y = 2(x - 5). Dividing by 2 instead of multiplying fails to undo the original division.

194. Solve the exponential equation $2^{x + 1} = 16$ for x.
     • 4
     • 8
     • 5
     • 3

     Correct answer: 3

     The solution is x = 3. Both sides can be written with base 2 since 16 equals $2^4$, so $2^{x + 1} = 2^4$ means the exponents are equal, giving x + 1 = 4 and therefore x = 3. Choosing 4 forgets to subtract the 1 from the exponent, but x + 1 must equal 4, so x is 3.

195. Use the power rule for logarithms to rewrite log base 2 of $x^5$.
     • 5 log base 2 of x
     • Log base 2 of x + 5
     • (log base 2 of x) raised to the 5th power
     • 5 + log base 2 of x

     Correct answer: 5 log base 2 of x

     The rewritten form is 5 log base 2 of x. The power rule lets an exponent inside a logarithm move to the front as a coefficient, so log base 2 of $x^5$ becomes 5 log base 2 of x. Raising the entire logarithm to the 5th power misapplies the rule, since the exponent multiplies the log rather than acting as an outer power.

196. A car loses value according to $V = 20000(0.8)^t$, where t is in years. What is the value after 1 year?
     • 18000
     • 16000
     • 4000
     • 10000

     Correct answer: 16000

     The value is 16000. Substituting t = 1 gives $V = 20000(0.8)^1$, and since 0.8 raised to the power 1 is just 0.8, this becomes 20000 times 0.8 = 16000. Subtracting 0.8 from 20000 or treating 0.8 as a flat loss gives a wrong figure, but the decay multiplies the value by 0.8 each year.

197. Evaluate the natural logarithm $\ln(e^4)$.
     • $e^4$
     • 4
     • 1
     • 4e

     Correct answer: 4

     The value is 4. The natural logarithm and the base e are inverse operations, so $\ln(e^4)$ simply returns the exponent, which is 4. Leaving the answer as $e^4$ ignores the cancellation, but ln of e raised to any power equals that power because ln has base e.

198. Condense the expression log base 3 of 5 + log base 3 of 4 into a single logarithm.
     • Log base 3 of 9
     • Log base 3 of 1.25
     • Log base 3 of 20
     • Log base 6 of 20

     Correct answer: Log base 3 of 20

     The condensed form is log base 3 of 20. The product rule states that the sum of two logarithms with the same base equals the log of the product, so log base 3 of 5 + log base 3 of 4 becomes log base 3 of (5 times 4) = log base 3 of 20. Adding the arguments to get 9 misuses the rule, since the arguments are multiplied, not added.

199. What is the value of log base 10 of 1000?
     • 3
     • 100
     • 10
     • 2

     Correct answer: 3

     The value is 3. The common logarithm log base 10 of 1000 asks for the power of 10 that gives 1000, and since 10 raised to the power 3 equals 1000, the answer is 3. Choosing 100 confuses the argument with the answer, but $10^3$ equals 1000 while $10^2$ equals only 100.

200. Given f(x) = 3x and g(x) = x + 2, find the composition (f composed with g)(x).
     • 3x + 2
     • 3x + 6
     • X + 6
     • $3x^2 + 2$

     Correct answer: 3x + 6

     The composition is 3x + 6. The notation (f composed with g)(x) means f(g(x)), so substitute g(x) = x + 2 into f to get f(x + 2) = 3(x + 2) = 3x + 6. Writing 3x + 2 only multiplies the x term and ignores distributing 3 across the entire input x + 2.

201. Solve the exponential equation $10^x = 0.01$ for x.
     • 2
     • -1
     • 0.1
     • -2

     Correct answer: -2

     The solution is x = -2. The value 0.01 equals 10 raised to the power negative 2, so $10^x = 10^{-2}$ means x = -2. Choosing positive 2 ignores that 0.01 is less than 1, which requires a negative exponent because 10 raised to a positive power gives a number greater than 1.

202. Solve the logarithmic equation log base 2 of (x + 3) = 4 for x.
     • 5
     • 13
     • 11
     • 1

     Correct answer: 13

     The solution is x = 13. Converting log base 2 of (x + 3) = 4 to exponential form gives 2 raised to the power 4 equals x + 3, so 16 = x + 3, and subtracting 3 gives x = 13. Forgetting to subtract the 3 leaves 16, but the argument x + 3 equals 16, so x must be 13.

203. What is the inverse function of the exponential function $f(x) = 7^x$?
     • F inverse (x) = log base 7 of x
     • F inverse (x) = $x^7$
     • F inverse (x) = log base x of 7
     • F inverse (x) = 7 / x

     Correct answer: F inverse (x) = log base 7 of x

     The inverse is f inverse (x) = log base 7 of x. The logarithm is defined as the inverse of an exponential function with the same base, so the inverse of $f(x) = 7^x$ is the logarithm base 7. Writing $x^7$ confuses an exponential function with a power function, but only a logarithm undoes a base raised to a variable exponent.

204. Evaluate log base 6 of 1.
     • 0
     • 1
     • 6
     • Undefined

     Correct answer: 0

     The value is 0. Any positive base raised to the power 0 equals 1, so log base 6 of 1 asks for the exponent that gives 1, which is 0. Choosing 1 confuses the argument with the answer, but $6^0$ equals 1 while $6^1$ equals 6.

205. A savings account follows the continuous growth model $A = 1000e^{0.05t}$. Which value represents the initial amount when t = 0?
     • 1050
     • 1005
     • 1000
     • 0

     Correct answer: 1000

     The initial amount is 1000. Setting t = 0 gives $A = 1000e^{0.05 \times 0} = 1000e^0$, and since $e^0$ equals 1, the result is 1000 times 1 = 1000. Choosing 1050 incorrectly adds growth at time zero, but no time has passed when t = 0, so the exponent is 0 and the account holds only the starting amount.

206. A triangle has a base of 14 centimeters and a height of 6 centimeters. What is its area?
     • 20 square centimeters
     • 42 square centimeters
     • 84 square centimeters
     • 42 centimeters

     Correct answer: 42 square centimeters

     The area is 42 square centimeters. The area of a triangle equals one-half times the base times the height, so A = (1/2) times 14 times 6 = (1/2) times 84 = 42. Choosing 84 square centimeters forgets the factor of one-half and gives the area of a rectangle with those dimensions instead of a triangle.

207. A circle has a radius of 7 inches. Using the formula C = 2 pi r, what is its circumference in terms of pi?
     • 7 pi inches
     • 14 pi inches
     • 49 pi inches
     • 28 pi inches

     Correct answer: 14 pi inches

     The circumference is 14 pi inches. The circumference of a circle equals 2 times pi times the radius, so C = 2 times pi times 7 = 14 pi. Choosing 49 pi inches squares the radius, which produces the area instead, because squaring is the operation used for area rather than for circumference.

208. A circle has a radius of 4 meters. Using the formula A = pi r squared, what is its area in terms of pi?
     • 8 pi square meters
     • 4 pi square meters
     • 16 pi square meters
     • 16 pi meters

     Correct answer: 16 pi square meters

     The area is 16 pi square meters. The area of a circle equals pi times the radius squared, so A = pi times 4 squared = pi times 16 = 16 pi. Choosing 8 pi square meters doubles the radius instead of squaring it, but the formula requires the radius to be multiplied by itself.

209. Two angles are complementary, and one of them measures 35 degrees. What is the measure of the other angle?
     • 145 degrees
     • 65 degrees
     • 55 degrees
     • 35 degrees

     Correct answer: 55 degrees

     The other angle is 55 degrees. Complementary angles add up to 90 degrees, so the missing angle is 90 minus 35 = 55. Choosing 145 degrees uses 180 instead of 90, which is the rule for supplementary angles rather than complementary angles.

210. Two angles are supplementary, and one of them measures 110 degrees. What is the measure of the other angle?
     • 70 degrees
     • 20 degrees
     • 250 degrees
     • 90 degrees

     Correct answer: 70 degrees

     The other angle is 70 degrees. Supplementary angles add up to 180 degrees, so the missing angle is 180 minus 110 = 70. Choosing 20 degrees uses 90 instead of 180, which is the rule for complementary angles rather than supplementary angles.

211. In a right triangle, the angle theta has an opposite side of length 8 and a hypotenuse of length 17. What is sin(theta)?
     • 17/8
     • 8/15
     • 15/17
     • 8/17

     Correct answer: 8/17

     The value is 8/17. The sine of an angle equals the length of the opposite side divided by the hypotenuse, so sin(theta) = 8/17. Choosing 17/8 inverts the ratio by placing the hypotenuse over the opposite side, which does not match the definition of sine.

212. A rectangular box has a length of 6 inches, a width of 3 inches, and a height of 4 inches. What is its volume?
     • 13 cubic inches
     • 72 cubic inches
     • 36 cubic inches
     • 72 square inches

     Correct answer: 72 cubic inches

     The volume is 72 cubic inches. The volume of a rectangular box equals length times width times height, so V = 6 times 3 times 4 = 72. Choosing 13 cubic inches adds the three dimensions instead of multiplying them, but volume requires the product of all three measurements.

213. The two acute angles of a right triangle are in the ratio 2 to 3. What is the measure of the larger acute angle?
     • 36 degrees
     • 54 degrees
     • 108 degrees
     • 60 degrees

     Correct answer: 54 degrees

     The larger acute angle is 54 degrees. The two acute angles of a right triangle add to 90 degrees, so with parts 2x and 3x we have 2x + 3x = 90, giving 5x = 90 and x = 18; the larger angle is 3 times 18 = 54. Choosing 36 degrees reports the smaller angle, 2 times 18, rather than the larger one that the question asks for.

214. A square has an area of 81 square feet. What is the length of one side?
     • 40.5 feet
     • 9 feet
     • 20.25 feet
     • 18 feet

     Correct answer: 9 feet

     The side length is 9 feet. The area of a square equals the side length squared, so the side equals $\sqrt{\text{area}}$: $\sqrt{81}$ is 9. Choosing 40.5 feet divides the area by 2, but finding a side from area requires taking a square root, not halving.

215. A central angle in a circle measures 90 degrees, and the circle has a radius of 6 inches. What fraction of the circle's full area does the resulting sector cover?
     • One-half
     • One-third
     • One-fourth
     • One-eighth

     Correct answer: One-fourth

     The sector covers one-fourth of the circle. A full circle is 360 degrees, so a 90-degree central angle covers 90/360, which simplifies to one-fourth. Choosing one-half would correspond to a 180-degree angle, which is twice the angle actually given.

216. In a right triangle, the side adjacent to angle theta has length 5 and the opposite side has length 12. What is tan(theta)?
     • 5/12
     • 13/12
     • 12/13
     • 12/5

     Correct answer: 12/5

     The value is 12/5. The tangent of an angle equals the length of the opposite side divided by the adjacent side, so tan(theta) = 12/5. Choosing 5/12 reverses the ratio by dividing the adjacent side by the opposite side, which contradicts the definition of tangent.

217. A trapezoid has parallel sides of length 10 and 14 and a height of 5. What is its area?
     • 60 square units
     • 120 square units
     • 35 square units
     • 350 square units

     Correct answer: 60 square units

     The area is 60 square units. The area of a trapezoid equals one-half times the sum of the parallel sides times the height, so A = (1/2) times (10 + 14) times 5 = (1/2) times 24 times 5 = 60. Choosing 120 square units omits the factor of one-half, doubling the correct result.

218. An angle measures 45 degrees. What is its equivalent measure in radians?
     • Pi/2 radians
     • Pi/3 radians
     • Pi/4 radians
     • Pi/6 radians

     Correct answer: Pi/4 radians

     The measure is pi/4 radians. To convert degrees to radians, multiply by pi over 180, so 45 times pi/180 = 45 pi/180, which simplifies to pi/4. Choosing pi/2 radians corresponds to 90 degrees, which is twice the given angle.

219. A sphere has a radius of 3 centimeters. Using the formula V = (4/3) pi r cubed, what is its volume in terms of pi?
     • 12 pi cubic centimeters
     • 36 pi cubic centimeters
     • 27 pi cubic centimeters
     • 108 pi cubic centimeters

     Correct answer: 36 pi cubic centimeters

     The volume is 36 pi cubic centimeters. The volume of a sphere equals four-thirds times pi times the radius cubed, so V = (4/3) times pi times 3 cubed = (4/3) times pi times 27 = 36 pi. Choosing 108 pi cubic centimeters forgets the four-thirds factor by not dividing by 3, since 4 times 27 equals 108 before the division.

220. Two parallel lines are cut by a transversal, and one of a pair of corresponding angles measures 73 degrees. What is the measure of the other corresponding angle?
     • 107 degrees
     • 17 degrees
     • 73 degrees
     • 146 degrees

     Correct answer: 73 degrees

     The other angle is 73 degrees. When two parallel lines are cut by a transversal, corresponding angles are equal, so the matching angle also measures 73 degrees. Choosing 107 degrees treats the angles as supplementary, but corresponding angles are congruent rather than adding to 180.

221. A regular hexagon has a perimeter of 48 centimeters. What is the length of one side?
     • 6 centimeters
     • 8 centimeters
     • 12 centimeters
     • 24 centimeters

     Correct answer: 8 centimeters

     Each side is 8 centimeters. A regular hexagon has six equal sides, so each side equals the perimeter divided by 6: 48 divided by 6 = 8. Choosing 12 centimeters divides by 4 as if the figure were a square, but a hexagon has six sides rather than four.

222. The interior angles of a triangle measure 40 degrees and 75 degrees. What is the measure of the third angle?
     • 115 degrees
     • 65 degrees
     • 45 degrees
     • 105 degrees

     Correct answer: 65 degrees

     The third angle is 65 degrees. The interior angles of any triangle add up to 180 degrees, so the third angle is 180 minus (40 + 75) = 180 minus 115 = 65. Choosing 115 degrees stops at the sum of the two known angles without subtracting from 180.

223. A right circular cone has a radius of 6 feet and a height of 10 feet. Using the formula V = (1/3) pi r squared h, what is its volume in terms of pi?
     • 360 pi cubic feet
     • 120 pi cubic feet
     • 60 pi cubic feet
     • 1080 pi cubic feet

     Correct answer: 120 pi cubic feet

     The volume is 120 pi cubic feet. The volume of a cone equals one-third times pi times the radius squared times the height, so V = (1/3) times pi times 6 squared times 10 = (1/3) times pi times 36 times 10 = (1/3) times 360 pi = 120 pi. Choosing 360 pi cubic feet omits the one-third factor, which is what distinguishes a cone from a cylinder of the same base and height.