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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 $-\frac{1}{2}$.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 $2^4 = 16$.

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.

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.

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.

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.

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.

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.

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.

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.

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 $+4x$.

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.

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.

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.

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.

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.

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.