GRE Quiz: Algebraic Expressions Simplification

What this quiz covers

This quiz focuses on Algebraic Expressions Simplification, giving you a quick way to practice the rules, question types, and explanations that matter most for GRE.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. $\dfrac{1}{2}x^2$
2. $0$ (correct answer)
3. $\dfrac{1}{4}x$
4. $\dfrac{1}{4}x^2$
5. $-\dfrac{1}{2}x^2$

Explanation: This question tests algebraic expression manipulation involving exponent rules and simplification. The correct approach is to expand $\left(\frac{1}{2}x\right)^2$ using the power rule: when squaring a product, square each factor. Thus $\left(\frac{1}{2}x\right)^2 = \left(\frac{1}{2}\right)^2 \cdot x^2 = \frac{1}{4}x^2$. Now we evaluate: $\frac{1}{4}x^2 - \frac{1}{4}x^2 = 0$ for all real $x$. The correct answer is B because these identical terms cancel completely. Choice D ($\frac{1}{4}x^2$) represents just the first term without the subtraction. Choice A ($\frac{1}{2}x^2$) might result from incorrectly squaring only the $x$ and not the coefficient $\frac{1}{2}$.

Question 2

1. $0$ (correct answer)
2. $6x$
3. $-18$
4. $2x^2-9$
5. $x^2-9$

Explanation: This question tests algebraic expression manipulation by recognizing equivalent forms of the same expression. The correct approach is to expand $(x-3)(x+3)$ using the difference of squares pattern: $(x-3)(x+3) = x^2 - 9$. Now we can substitute this into the original expression: $x^2 - 9 - (x^2 - 9) = x^2 - 9 - x^2 + 9 = 0$. The correct answer is A because the two identical expressions cancel completely. Choice C ($-18$) might result from incorrectly thinking we get $-9 - 9 = -18$. Choice E ($x^2 - 9$) represents just the first part of the expression without considering the subtraction.

Question 3

1. $\dfrac{x+3}{x-3}$
2. $x-3$
3. $x^2-9$
4. $x+3$ (correct answer)
5. $x+6$

Explanation: This question tests algebraic expression manipulation. The correct simplification strategy is to factor the numerator as a difference of squares and cancel the common factor with the denominator. The numerator factors to (x - 3)(x + 3). Canceling the (x - 3) term with the denominator for x ≠ 3 yields x + 3. This matches the correct answer of x + 3. A representative distractor like x - 3 fails due to incorrectly factoring or subtracting instead of adding. Another error could involve not canceling and leaving it as x^2 - 9.

Question 4

1. $6-w$
2. $6+2w$
3. $\dfrac{2}{3}9-3w$
4. $6-2w$ (correct answer)
5. $\dfrac{18-3w}{3}$

Explanation: This question tests algebraic expression manipulation. The correct simplification strategy is to distribute the 2/3 through the parentheses. Distribute to 9 to get 6, and to -3w to get -2w. The result is 6 - 2w. This matches the correct answer of 6 - 2w. A representative distractor like 6 + 2w fails due to a sign error in distribution. Another mistake could be misapplying the fraction, leading to 6 - w.

Question 5

1. $a-b$
2. $a+b$ (correct answer)
3. $\dfrac{a^2-b^2}{a}$
4. $a^2-b^2$
5. $\dfrac{a+b}{a-b}$

Explanation: This question tests algebraic expression manipulation involving factoring and simplification of rational expressions. The correct simplification strategy is to recognize that the numerator $a^2 - b^2$ is a difference of squares that factors as $(a+b)(a-b)$. Substituting this factorization gives us $\frac{(a+b)(a-b)}{a-b}$. Since $a \neq b$, we know that $a-b \neq 0$, so we can cancel the common factor $(a-b)$ from numerator and denominator. This leaves us with $a+b$, which is choice B. Choice A ($a-b$) might result from incorrectly thinking the expression simplifies to the denominator. Choice D ($a^2-b^2$) represents the numerator alone without considering the division.

Question 6

1. $-16$
2. $4$
3. $7$
4. $16$ (correct answer)
5. $(-3)^2-2k+1$

Explanation: This question tests algebraic expression manipulation. The correct evaluation strategy is to substitute k = -3 into the expression and compute. Substitute to get (-3)^2 - 2(-3) + 1. Compute 9 + 6 + 1 = 16. This matches the correct answer of 16. A representative distractor like 4 fails due to a sign error in the linear term. Another mistake could be incorrect exponentiation, leading to 7.

Question 7

1. $5-10z+4z$
2. $5+6z$
3. $9-2z$
4. $5-6z$ (correct answer)
5. $5(1-2z+4z)$

Explanation: This question tests algebraic expression manipulation. The correct simplification strategy is to distribute the 5 and then combine like terms with the +4z. Distribute 5 to get 5 - 10z. Add 4z to get 5 - 10z + 4z = 5 - 6z. This matches the correct answer of 5 - 6z. A representative distractor like 5 + 6z fails due to a sign error in distribution. Another common mistake is incorrect combining, leading to 9 - 2z.

Question 8

1. $\dfrac{x}{x+1}$
2. $x+1$
3. $x^2+1$
4. $x$ (correct answer)
5. $(x+1)^2-(x+1)$

Explanation: This question tests algebraic expression manipulation. The correct simplification strategy is to factor the numerator and cancel the common factor. Factor (x + 1)^2 - (x + 1) as (x + 1)(x + 1 - 1) = (x + 1)x. Cancel (x + 1) with the denominator for x ≠ -1, leaving x. This matches the correct answer of x. A representative distractor like x + 1 fails due to incomplete factoring. Another error is not subtracting correctly, leading to x^2 + 1.

Question 9

1. $\dfrac{x}{4}$
2. $-\dfrac{x}{4}$ (correct answer)
3. $-\dfrac{2x}{4}$
4. $\dfrac{4x}{8}$
5. $\dfrac{x-3x}{4}$

Explanation: This question tests algebraic expression manipulation. The correct simplification strategy is to find a common denominator and combine the fractions. The common denominator is 4, so rewrite as (2x / 4) - (3x / 4). Combine to (2x - 3x) / 4 = -x / 4. This matches the correct answer of -x / 4. A representative distractor like x / 4 fails due to a sign error in subtraction. Another mistake could be adding denominators, leading to something like 4x / 8.

Question 10

1. $\dfrac{x-2}{x+2}$
2. $x+2$
3. $x^2-4$
4. $x-2$ (correct answer)
5. $(x-2)(x+2)$

Explanation: This question tests algebraic expression manipulation. The correct simplification strategy is to factor the numerator and cancel the common factor with the denominator. The numerator is (x - 2)(x + 2). Cancel (x + 2) with the denominator for x ≠ -2, leaving x - 2. This matches the correct answer of x - 2. A representative distractor like x + 2 fails due to a sign error in cancellation. Another mistake is not simplifying, leaving (x - 2)/(x + 2).

Question 11

1. $\dfrac{x-1}{x}$
2. $\dfrac{3x}{x-1}$
3. $\dfrac{x}{1-x}$
4. $\dfrac{x}{x-1}$ (correct answer)
5. $\dfrac{x}{x+1}$

Explanation: This question tests algebraic expression manipulation. The correct simplification strategy is to combine the fractions over a common denominator and simplify the numerator. Both terms have the denominator x - 1, so subtract the numerators: 2x - x = x. The result is x / (x - 1) for x ≠ 1. This matches the correct answer of x / (x - 1). A representative distractor like 3x / (x - 1) fails due to adding instead of subtracting the numerators. Another error could be inverting the sign, leading to x / (1 - x).

Question 12

1. $m+10$
2. $-m+10$ (correct answer)
3. $m-10$
4. $5m+4$
5. $4m-2m-7+3- m$

Explanation: This question tests algebraic expression manipulation. The correct simplification strategy is to distribute the negative and the 3, then combine like terms. Distribute the negative to get 4m - 2m + 7, and distribute 3 to get 3 - 3m. Combine 4m - 2m - 3m = -m and 7 + 3 = 10, resulting in -m + 10. This matches the correct answer of -m + 10. A representative distractor like m + 10 fails due to a sign error in distributing the negative. Another common mistake is mishandling the 3(1 - m), leading to m - 10.

Question 13

1. $8x-16$
2. $-8x+16$
3. $-8(x-4)$ (correct answer)
4. $8(x-4)$
5. $(x-4)\big((x-4)-(x+4)\big)$

Explanation: This question tests algebraic expression manipulation. The correct simplification strategy is to expand the terms and combine like terms, or recognize factoring opportunities. Expand (x - 4)^2 to x^2 - 8x + 16 and (x - 4)(x + 4) to x^2 - 16, then subtract: x^2 - 8x + 16 - x^2 + 16 = -8x + 32. Factor to -8(x - 4). This matches the correct answer of -8(x - 4). A representative distractor like -8x + 16 fails due to a sign error in subtraction. Another mistake is not distributing the negative, leading to 8x - 16.

Question 14

1. $x(x-4)$
2. $x+4$
3. $\dfrac{x-4}{x}$
4. $x-4$ (correct answer)
5. $\dfrac{x^2}{x}-4x$

Explanation: This question tests algebraic expression manipulation. The correct simplification strategy is to factor the numerator and cancel the common factor with the denominator. Factor x^2 - 4x to x(x - 4). Cancel the x in the denominator for x ≠ 0, leaving x - 4. This matches the correct answer of x - 4. A representative distractor like x + 4 fails due to a sign error in factoring. Another mistake could be not canceling, leading to (x - 4)/x.

Question 15

1. $4x-7$
2. $4x-23$ (correct answer)
3. $8x-23$
4. $4x+23$
5. $6x-2(x+4)$

Explanation: This question tests algebraic expression manipulation. The correct simplification strategy is to distribute the coefficients through the parentheses and then combine like terms. First, distribute the 3 to get 6x - 15. Next, distribute the -2 to get -2x - 8, then combine the like terms: 6x - 2x = 4x and -15 - 8 = -23. This matches the correct answer of 4x - 23. A representative distractor like 4x - 7 fails due to a sign error, perhaps by adding the constants instead of combining them properly with the negative sign. Another common mistake is forgetting the negative sign on the second parenthesis, leading to 4x + 23.

Question 16

1. $a(2a-3)+5$ (correct answer)
2. $2a(a-3)+5$
3. $2a^2-3(a+5)$
4. $(2a^2-3a)+5a$
5. $2a^2-(3a+5)$

Explanation: This question tests algebraic expression manipulation by asking which expression simplifies to $2a^2-3a+5$. The correct strategy is to expand each choice and check which matches the target expression. For choice A: $a(2a-3)+5 = 2a^2 - 3a + 5$, which exactly matches our target. For choice B: $2a(a-3)+5 = 2a^2 - 6a + 5$, which has the wrong coefficient for the middle term. The correct answer is A because it expands to match the given expression perfectly. A common error would be to incorrectly distribute, such as thinking $a(2a-3) = 2a^2 - 3$ (forgetting to multiply the $-3$ by $a$), which would lead to an incorrect simplification.

Question 17

1. $\dfrac{6}{2x}$
2. $\dfrac{8}{2x}$
3. $\dfrac{8x}{x}$
4. $\dfrac{8}{x}$ (correct answer)
5. $\dfrac{8}{x^2}$

Explanation: This question tests algebraic expression manipulation. The correct simplification strategy is to combine the fractions over a common denominator. Both terms have denominator x, so add the numerators: 6 + 2 = 8. The result is 8 / x for x ≠ 0. This matches the correct answer of 8 / x. A representative distractor like 8 / 2x fails due to incorrectly combining denominators. Another error could be multiplying numerators, leading to 12 / x^2 or similar.

Question 18

1. $\dfrac{a^2}{b^2}$
2. $\dfrac{b}{a}$
3. $ab$
4. $\dfrac{a}{b}$ (correct answer)
5. $\dfrac{a^2b}{ab}$

Explanation: This question tests algebraic expression manipulation. The correct evaluation strategy is to cancel common factors in the numerator and denominator. Cancel one a from numerator and denominator, and cancel b from numerator and one b in denominator. This simplifies to a / b for a ≠ 0 and b ≠ 0. This matches the correct answer of a / b. A representative distractor like b / a fails due to inverting the variables incorrectly. Another error could be squaring terms, leading to a^2 / b^2.

Question 19

1. $x-\dfrac{4}{x}$ (correct answer)
2. $x-4$
3. $\dfrac{x^2-4}{x}$
4. $\dfrac{2x^2-8}{x}$
5. $\dfrac{2(x-4)}{2x}$

Explanation: This question tests algebraic expression simplification of rational expressions. The correct strategy is to factor the numerator and simplify by canceling common factors. We have $\frac{2x^2-8}{2x} = \frac{2(x^2-4)}{2x} = \frac{x^2-4}{x}$. This can be written as $\frac{x^2}{x} - \frac{4}{x} = x - \frac{4}{x}$. The Aorrect answer matches this simplified form of $x - \frac{4}{x}$ (choice A). A common error would be to incorrectly cancel terms instead of factors, such as thinking $\frac{x^2-4}{x} = x-4$ (choice B), which ignores that subtraction doesn't allow term-by-term division.

Question 20

1. $4x-23$ (correct answer)
2. $4x-7$
3. $6x-15-2x+4$
4. $8x-23$
5. $4x+23$

Explanation: This question tests algebraic expression manipulation through distribution and combining like terms. The correct simplification strategy is to distribute the multiplication over each parenthetical expression, then combine like terms. First, distribute: $3(2x-5) = 6x - 15$ and $-2(x+4) = -2x - 8$. Next, combine these results: $6x - 15 - 2x - 8 = 6x - 2x - 15 - 8 = 4x - 23$. The correct answer is A because it matches this simplified form. Choice D ($8x - 23$) represents a common error where students might incorrectly add the coefficients of $x$ instead of subtracting them. Choice E ($4x + 23$) shows a sign error in combining the constant terms.