20 medium SAT Equivalent expressions questions

Medium is where most scores are actually won and lost. These questions are not tricky for the sake of it, but every one of them has a wrong answer built to catch a specific shortcut.

Every question below is a real item from the SAT Climb bank, tagged medium by the same difficulty model the app uses to build your practice. Pick an answer before you open the explanation.

Math · Advanced Math~3 per testMedium tier

None of these repeat the 25 hard Equivalent expressions questions, so the two pages together are 45 different questions on this one skill.

Question 1Medium

Which expression is equivalent to 3x2+14x+83x^{2} + 14x + 8?

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Why B is right

To factor 3x2+14x+83x^{2} + 14x + 8, find two numbers that multiply to 3·8 = 24 and add to 14. These are 12 and 2. Rewrite as 3x2+12x+2x+83x^{2} + 12x + 2x + 8, then factor by grouping: 3x(x+4)+2(x+4)=(3x+2)(x+4)3x(x + 4) + 2 (x + 4) = (3x + 2) (x + 4).

Why the others are wrong

  • AThis expands to 3x² + 10x + 8, which has the wrong middle coefficient.
  • CThis results from incorrect signs; expanding gives 3x² - 14x + 8 instead of the original expression.
  • DThis expands to 3x² + 11x + 8, which has the wrong middle coefficient.
Question 2Medium

Which expression is equivalent to 6x3−13x2−5x6x^{3} - 13x^{2} - 5x?

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Why B is right

First factor out x: x(6x2−13x−5)x(6x^{2} - 13x - 5). Then factor the quadratic 6x2−13x−5=(3x+1)(2x−5)6x^{2} - 13x - 5 = (3x + 1) (2x - 5). Therefore the complete factorization is x(3x+1)(2x−5)x(3x + 1) (2x - 5).

Why the others are wrong

  • AThis reverses the order of the two binomial factors, which doesn't affect the product but may confuse the matching pattern.
  • CThis uses incorrect signs in both binomial factors; expanding gives x(6x² + 13x - 5), not x(6x² - 13x - 5).
  • DThis uses -4 instead of -5 in the second factor; expanding gives x(6x² - 10x - 4), not x(6x² - 13x - 5).
Question 3Medium

Which expression is equivalent to (x4y6)12(x3y2)12\displaystyle (x^{4}y^{6})^{\frac{1}{2}}(x^{3}y^{2})^{\frac{1}{2}}, where x and y are positive?

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Why B is right

Using the power rule (am)n=amn(a^{m})^{n} = a^{mn}, (x4y6)12\displaystyle (x^{4}y^{6}) ^\frac{1}{2} = x2y3x^{2} y^{3} and (x3y2)(x^{3} y^{2})^12=[MATH]x32\displaystyle \frac{1}{2} = [MATH]x^\frac{3}{2}[/MATH]y. Multiplying gives x2x^{2}·[MATH]x32\displaystyle [MATH]x^\frac{3}{2}[/MATH]·y3y^{3}·y=[MATH]x(2+3/2)y = [MATH]x^(2+3/2)\cdot y(3+1)y^(3+1) = x72\displaystyle x^\frac{7}{2}[/MATH]y⁴.

Why the others are wrong

  • AThis results from incorrectly computing the exponents, rounding 7/2 down to 3 or making errors in adding fractional exponents.
  • CThis results from failing to apply the power rule correctly, multiplying exponents instead of applying the 1/2 power, treating it as if the exponents were simply added.
  • DThis results from incorrectly computing the exponent of y, treating 3 + 1 as 3 or making an error in adding the exponents of y.
Question 4Medium

Which expression is equivalent to 8x3y62xy2\displaystyle \frac{8x^{3}y^{6}}{2xy^{2}}?

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Why B is right

Dividing coefficients gives 8/2=48/2 = 4. For exponents, x3x^{3}/x = x2x^{2} and y6y^{6}/y2y^{2} = y4y^{4}. The simplified expression is 4x2y44x^{2}y^{4}.

Why the others are wrong

  • AThis incorrectly computes y⁶/y² as y³ instead of y⁴.
  • CThis fails to subtract exponents for x, keeping x³ instead of reducing to x².
  • DThis uses 6 as the coefficient instead of correctly dividing 8 by 2.
Question 5Medium

Which expression is equivalent to 2x2+11x+122x^{2} + 11x + 12?

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Why A is right

Factoring 2x2+11x+122x^{2} + 11x + 12 requires finding two binomials whose product gives the original expression. The factors (2x+3)(x+4)(2x + 3) (x + 4) expand to 2x2+8x+3x+122x^{2} + 8x + 3x + 12, which equals 2x2+11x+122x^{2} + 11x + 12.

Why the others are wrong

  • BThis choice produces 2x² + 10x + 12 when expanded, giving an incorrect middle term coefficient.
  • CThis choice has sign errors in both binomials, yielding 2x² - 11x + 12 instead of the original expression.
  • DThis choice produces 2x² + 14x + 12 when expanded, with an incorrect middle term.
Question 6Medium

Which expression is equivalent to (8m6n3)23\displaystyle (8m^{6}n^{3})^{\frac{2}{3}}, where m and n are positive?

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Why A is right

For positive values, (abc)(abc)^k=ak⋅bk⋅ckk = a^k\cdot b^k\cdot c^k. Therefore, (8m6n3)(8m^{6} n^{3})^23=[MATH]823\displaystyle \frac{2}{3} = [MATH]8^\frac{2}{3}\cdot m(6⋅2/3)m^(6\cdot 2/3)\cdot n(3⋅2/3)n^(3\cdot 2/3)[/MATH]. Since [MATH]823\displaystyle [MATH]8^\frac{2}{3} = (813)\displaystyle (8^\frac{1}{3}) ^{2}[/MATH] = 22=42^{2} = 4, this simplifies to 4m4n24m^{4} n^{2}.

Why the others are wrong

  • BThis results from incorrectly computing 8^(2/3) as 2 instead of 4.
  • CThis results from adding 2/3 to the exponents instead of multiplying by 2/3.
  • DThis results from incorrectly computing 3·2/3 as 1 instead of 2.
Question 7Medium

Which expression is equivalent to x4−16x2+4\displaystyle \frac{x^{4} - 16}{x^{2} + 4}?

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Why A is right

The numerator x⁴ - 16 is a difference of squares that factors as (x2x^{2} - 4)(x2x^{2} + 4). The expression becomes (x2x^{2} - 4)(x2x^{2} + 4) / (x2x^{2} + 4). The factor (x2x^{2} + 4) cancels from numerator and denominator, leaving x2x^{2} - 4.

Why the others are wrong

  • BThis results from incorrectly canceling the x² terms instead of properly factoring the numerator and canceling common factors.
  • CThis results from incorrectly simplifying 16/4 as 8 instead of recognizing the difference of squares pattern in the numerator.
  • DThis results from a sign error when factoring or canceling, incorrectly treating the expression as if it had a negative leading coefficient.
Question 8Medium

Which expression is equivalent to 48x5y8\displaystyle \sqrt{48x^{5}y^{8}}, where x and y are positive?

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Why A is right

Since 48=16⋅348 = 16\cdot 3, x5=x4⋅xx^{5} = x^{4}\cdot x, and y8=(y4)2y^{8} = (y^{4}) ^{2}, we have 48x5y8=16⋅3⋅x4⋅x⋅y8=16⋅x4⋅y8⋅3x\displaystyle \sqrt{48x⁵y⁸} = \sqrt{16\cdot 3\cdot x⁴\cdot x\cdot y⁸} = \sqrt{16}\cdot \sqrt{x⁴}\cdot \sqrt{y⁸}\cdot \sqrt{3x} = 4x2y43x\displaystyle 4x^{2}y^{4} \sqrt{3x}.

Why the others are wrong

  • BThis results from incorrectly simplifying √(x⁵), extracting x² but failing to leave x under the radical, treating it as if x⁵ were a perfect square.
  • CThis results from incorrectly computing √48 as 12 instead of 4√3, or confusing the square root with a different operation.
  • DThis results from incorrectly simplifying √(x⁵) as x instead of x², extracting only one factor of x from x⁴·x.
Question 9Medium

Which expression is equivalent to (2x3y4)3(3x2y)(2x^{3}y^{4})^{3}(3x^{2}y)?

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Why B is right

Using the power rule (am)n=amn(a^{m})^{n} = a^{mn}, (2x3y4)3(2x^{3}y^{4}) ^{3} = 2³x⁹y¹2^{2}. Multiplying by 3x2y3x^{2}y gives 23⋅3⋅x2^{3}\cdot 3\cdot x⁹⁺2^{2}·y¹2^{2}⁺¹ = 8⋅3⋅x8\cdot 3\cdot x¹¹y¹3^{3} = 24x¹¹y¹3^{3}.

Why the others are wrong

  • AThis results from failing to add exponents when multiplying x⁹ by x², keeping the exponent at 9 instead of 11, and similarly for y.
  • CThis results from incorrectly applying the power rule to the exponent of x, computing 3 + 2 = 5 and then adding 2, or other exponent errors.
  • DThis results from computing the coefficient incorrectly as 2·3 = 6 instead of 2³·3 = 24.
Question 10Medium

Which expression is equivalent to 481x12y8\displaystyle ^{4}\sqrt{81x^{12}y^{8}}, where x and y are positive?

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Why A is right

The fourth root can be rewritten as 81x12y84\displaystyle \sqrt[4]{81x^{12}y^{8}} = ⁴81\displaystyle \sqrt{81} · ⁴x\displaystyle \sqrt{x}¹2^{2} · ⁴y8=3x3y2\displaystyle \sqrt{y}⁸ = 3x^{3} y^{2}, since [MATH]8114\displaystyle [MATH]81^\frac{1}{4} = 3[/MATH], [MATH]x124\displaystyle [MATH]x^\frac{12}{4}[/MATH] = x3x^{3}, and [MATH]y84\displaystyle [MATH]y^\frac{8}{4}[/MATH] = y2y^{2}.

Why the others are wrong

  • BThis incorrectly computes ⁴√81 = 9 instead of 3.
  • CThis makes errors in the exponents: 12/4 = 3 not 4, and 8/4 = 2 not 3.
  • DThis incorrectly computes 8/4 = 4 instead of 2 for the y exponent.
Question 11Medium

Which expression is equivalent to x4−81x+3\displaystyle \frac{x^{4} - 81}{x + 3}, where x≠−3x \ne -3?

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Why C is right

The numerator x⁴ - 81 factors as a difference of squares: (x2x^{2} - 9)(x2x^{2} + 9). The term x2x^{2} - 9 further factors as (x−3)(x+3)(x - 3) (x + 3). After canceling the common factor (x+3)(x + 3), the expression simplifies to (x−3)(x - 3)(x2x^{2} + 9).

Why the others are wrong

  • AThis results from attempting polynomial long division but making computational errors in the quotient terms.
  • BThis results from failing to cancel the correct factor; (x + 3) should have been canceled from numerator and denominator.
  • DThis results from a sign error in the second factor; x² + 9 was incorrectly written as x² - 9.
Question 12Medium

Which expression is equivalent to x−4y2z6x2y−3z4\displaystyle \frac{x^{-4}y^{2}z^{6}}{x^{2}y^{-3}z^{4}}, where x, y, and z are positive?

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Why A is right

Using the quotient rule for exponents: x⁻⁴/x2x^{2} = x⁻⁴⁻2^{2} = x⁻⁶, y2y^{2}/y⁻3^{3} = y2y^{2}⁻⁽⁻3^{3}⁾ = y⁵, and z⁶/z⁴ = z⁶⁻⁴ = z2z^{2}. Therefore, the expression equals x⁻⁶y⁵z2z^{2} = y⁵z2z^{2}/x⁶.

Why the others are wrong

  • BThis choice incorrectly calculates the exponent of x as -2 instead of -6.
  • CThis choice results from adding exponents instead of subtracting for z, giving z¹⁰ instead of z².
  • DThis choice contains an arithmetic error in the exponent of y, calculating 4 instead of 5.
Question 13Medium

Which expression represents the product of x−4y2z3x^{-4}y^{2}z^{3} and x6z2+y4z−5x^{6}z^{2} + y^{4}z^{-5}?

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Why B is right

Applying the distributive property yields x⁻⁴y2z3y^{2} z^{3}(x⁶z2z^{2}) + x⁻⁴y2z3y^{2} z^{3}(y⁴z⁻⁵). Using aᵐ·aⁿ = aᵐ⁺ⁿ, this simplifies to x⁻⁴⁺⁶y2z3y^{2} z^{3}⁺2^{2} + x⁻⁴y2y^{2}⁺⁴z3z^{3}⁻⁵, which equals x2y2x^{2} y^{2}z⁵ + x⁻⁴y⁶z⁻2^{2}.

Why the others are wrong

  • AThis results from failing to include the y² factor in the first term and the x⁻⁴ factor in the second term.
  • CThis results from omitting all y factors from both terms.
  • DThis results from multiplying the x exponents instead of adding them in the first term.
Question 14Medium

Which expression is equivalent to 8x6y33\displaystyle \sqrt[3]{8x^{6}y^{3}}, where x and y are positive?

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Why B is right

Apply the cube root to each factor: 38=2\displaystyle ^{3} \sqrt{8} = 2, 3x6=x\displaystyle ^{3} \sqrt{x⁶} = x⁶/3^{3} = x2x^{2}, and 3^{3}√(y3)(y^{3}) = y3y^{3}/3^{3} = y. Therefore, 3^{3}√(8x6y3)(8x^{6} y^{3}) = 2x2y2x^{2}y.

Why the others are wrong

  • AThis choice incorrectly applies the cube root to x⁶, calculating x³ instead of x².
  • CThis choice contains an error in the exponent of y, calculating y² instead of y.
  • DThis choice fails to apply the cube root to the coefficient 8, leaving it unchanged.
Question 15Medium

Which expression is equivalent to 8x3+278x^{3} + 27?

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Why A is right

The expression 8x3+278x^{3} + 27 is a sum of cubes with a=2xa = 2x and b=3b = 3. Using the formula a3a^{3} + b3b^{3} = (a+b)(a2−ab+[MATH]b2)(a + b) (a^{2} - ab + [MATH]b^{2})[/MATH], this factors as (2x+3)(4x2−6x+9)(2x + 3) (4x^{2} - 6x + 9).

Why the others are wrong

  • BThis results from a sign error in the first factor; the sum of cubes requires (a + b), not (a - b).
  • CThis results from sign errors in the second factor; the middle term should be negative and the formula is a² - ab + b², not a² + ab + b².
  • DThis results from an error in computing b² = 9 in the last term of the second factor, incorrectly using 3 instead.
Question 16Medium

Which expression is equivalent to x8y124\displaystyle \frac{x^{8}y^{12}}{^{4}}, where x and y are positive?

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Why A is right

For positive values, (xayb)(x^{a}y^{b})^(1/n)(1/n) = x^(a/n)(a/n)y^(b/n)(b/n). Therefore, x8x^{8}y¹2^{2}/⁴ equals [MATH]x84\displaystyle [MATH]x^\frac{8}{4} y124\displaystyle y^\frac{12}{4}[/MATH], which simplifies to x2y3x^{2} y^{3}.

Why the others are wrong

  • BThis choice results from incorrectly halving the exponents instead of dividing by 4.
  • CThis choice correctly computes the x exponent but incorrectly divides 12 by 3 instead of 4 for y.
  • DThis choice results from failing to apply the root operation, leaving the exponents unchanged.
Question 17Medium

Which expression is equivalent to 3x2+11x−4x+4\displaystyle \frac{3x^{2} + 11x - 4}{x + 4}?

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Why C is right

The numerator 3x2+11x−43x^{2} + 11x - 4 factors as (3x−1)(x+4)(3x - 1) (x + 4). The expression becomes (3x−1)(x+4)/(x+4)(3x - 1) (x + 4) / (x + 4). The factor (x+4)(x + 4) cancels from numerator and denominator, leaving 3x−13x - 1.

Why the others are wrong

  • AThis results from a sign error when factoring the quadratic, incorrectly factoring as (3x + 1)(x + 4) instead of (3x - 1)(x + 4).
  • BThis results from an error in the constant term when factoring, incorrectly determining the factors of -4 in relation to the middle term.
  • DThis results from incorrectly believing that x + 4 remains in the simplified form, failing to recognize it cancels completely.
Question 18Medium

Which expression represents the product of (m2n4p3)(m^{2}n^{4}p^{3}) and (m3p5+n6p−2)(m^{3}p^{5} + n^{6}p^{-2})?

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Why B is right

Applying the distributive property, (m2(m^{2}n⁴p3)p^{3})(m3m^{3}p⁵ + n⁶p⁻2)^{2}) = m2m^{2}n⁴p3p^{3}(m3m^{3}p⁵) + m2m^{2}n⁴p3p^{3}(n⁶p⁻2)^{2}) = m⁵n⁴p⁸ + m2m^{2}n¹⁰p.

Why the others are wrong

  • AThis results from failing to carry the n⁴ factor through both terms of the distributive property.
  • CThis results from failing to carry the m² factor through to the second term.
  • DThis results from an error in adding the exponents of p in the second term, incorrectly calculating 3 + (-2) as -6 instead of 1.
Question 19Medium

Which expression is equivalent to (x4y6)3(x2y)2(x^{4}y^{6})^{3}(x^{2}y)^{2}?

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Why C is right

Using the power rule, (x4y6)3(x^{4}y^{6}) ^{3} = x12y18x^{12}y^{18} and ([MATH]x2y)([MATH]x^{2}y)^{2} = x[/MATH]⁴y2y^{2}. Multiplying these expressions using the product rule gives x¹2^{2}⁺⁴y¹⁸⁺2^{2} = x16y2x^{16} y^{2}⁰.

Why the others are wrong

  • AThis choice incorrectly adds exponents before applying the power rule, likely computing 4+2 then cubing.
  • BThis choice uses an incorrect x exponent, possibly from multiplying exponents incorrectly.
  • DThis choice uses an incorrect y exponent, possibly from failing to multiply 6 by 3 correctly.
Question 20Medium

Which expression is equivalent to 6x3y43x2y2\displaystyle \frac{6x^{3}y^{4}}{3x^{2}y^{2}}, where x≠0x \neq 0 and y≠0y \neq 0?

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Why D is right

Dividing coefficients: 6/3 = 2. Using the quotient rule for exponents: x3x^{3}/x2x^{2} = [MATH]x(3−2)[MATH]x^(3-2) = x[/MATH] and y⁴/y2y^{2} = [MATH]y(4−2)[MATH]y^(4-2)[/MATH] = y2y^{2}. Therefore, the expression simplifies to 2xy22xy^{2}.

Why the others are wrong

  • AThis results from incorrectly computing y⁴/y² as y instead of y².
  • BThis results from incorrectly adding exponents instead of subtracting them when dividing.
  • CThis results from incorrectly simplifying the coefficient 6/3 as 3 instead of 2.

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