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Math: Advanced Math, 3 original questions

SAT Equivalent Expressions Practice: Questions and Answers

Practice factoring, exponent rules and equivalent algebraic forms with three original SAT-style questions, complete answers and explanations of each choice.

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The short answer

Equivalent expressions have the same value for every input in their shared domain. Look for a common factor, a useful polynomial pattern or an exponent rule before doing more arithmetic. The best form depends on the task: factored form can expose zeros, while expanded form can expose coefficients. Keep restrictions on denominators and even roots. After rewriting, expand or substitute a permitted value to catch a mistake; one matching value alone does not prove equivalence.

Method at a glance

  1. 1Choose the form you need
  2. 2Apply one valid rule at a time
  3. 3Carry the domain restrictions
  4. 4Check the transformation

Try the 3 questionsRead the full method

Step-by-step method

Use these steps in order, then apply them to the questions below.

  1. Step 1: Choose the form you need

    Decide whether the question needs a factor, coefficient, zero or numerical value. Rewrite toward that target rather than expanding automatically.

  2. Step 2: Apply one valid rule at a time

    Factor every term, distribute to every term, or combine powers of the same base. For division, subtract exponents rather than dividing them.

  3. Step 3: Carry the domain restrictions

    A canceled denominator was still required to be nonzero in the original expression. Keep that restriction when comparing forms.

  4. Step 4: Check the transformation

    Expand factored expressions or check several permitted inputs to detect errors. Use algebraic rules, not isolated numerical agreement, to establish equivalence.

Try 3 SAT-style questions

These are original SAT-style questions written by PeakSAT, not official College Board questions. Choose an answer and check it to see why each choice is right or wrong, or open the worked answer.

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Question 1 of 3

Which expression is equivalent to 3x² − 12x for every real x?

Choose an answer for question 1
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Correct answer

Choice A: 3x(x − 4)

Explanation

Both terms contain 3x. Dividing 3x² by 3x gives x, and dividing −12x by 3x gives −4, so the expression is 3x(x − 4). Expanding gives 3x² − 12x again.

Why each choice is right or wrong

  1. Choice A

    Correct answer

    Correct: distributing 3x produces both original terms.

  2. Choice B

    This expands to 3x² − 12, losing the x in the second term.

  3. Choice C

    This expands to 3x² − 36x; the second term is too large.

  4. Choice D

    This expands to 3x² − 4x, so it does not preserve the coefficient −12.

Question 2 of 3

For nonzero x and y, which expression is equivalent to (4x³y²)/(2xy)?

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Correct answer

Choice B: 2x²y

Explanation

Divide the coefficients: 4/2 = 2. Subtract the exponents of each matching base: x³/x = x² and y²/y = y. The simplified expression is 2x²y, with x and y still nonzero as stated.

Why each choice is right or wrong

  1. Choice A

    The x exponent must decrease from 3 to 2 when divided by x.

  2. Choice B

    Correct answer

    Correct: 2x²y includes both coefficient division and exponent subtraction.

  3. Choice C

    The y exponent must decrease from 2 to 1 when divided by y.

  4. Choice D

    The x exponent is 3 − 1 = 2, not 1.

Question 3 of 3

What is the value of 9^(3/2)?

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Correct answer

Choice C: 27

Explanation

For a positive base, 9^(3/2) = (√9)³. The principal square root of 9 is 3, and 3³ = 27. Equivalently, √(9³) = √729 = 27.

Why each choice is right or wrong

  1. Choice A

    This multiplies 9 by 3/2 instead of applying an exponent.

  2. Choice B

    This does not perform the required square root and cube.

  3. Choice C

    Correct answer

    Correct: (√9)³ = 3³ = 27.

  4. Choice D

    This calculates 9³ but omits the square root.

Common mistakes

  • Factoring out 3x from only the first term. Divide every term by the proposed common factor.

  • Subtracting coefficients or exponents without matching the operation. Multiplication adds exponents; division subtracts them for the same base.

  • Treating a fractional exponent as multiplication. An exponent of 3/2 means square root followed by cubing.

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Common questions

Can I test one value to prove two expressions are equivalent?

No. Different expressions can agree at one input. Establish the identity with algebra; substitution is useful for finding an error.

What happens to a canceled denominator restriction?

It remains part of the original domain. Simplifying a fraction does not make an originally forbidden input valid.

When should I factor instead of expand?

Factor when zeros or multiplicative structure help. Expand when coefficients or combining like terms are the target. Choose the form that exposes the requested information.

Sources

  1. College Board: Advanced Mathsatsuite.collegeboard.org
  2. College Board: Student Question Bank Math skillssatsuite.collegeboard.org

Sources are linked for reference; they do not endorse PeakSAT. How we prepare our guides