Math · Advanced Math

Equivalent Expressions on the Digital SAT

Equivalent Expressions questions ask you to rewrite an algebraic expression in a different form without changing its value. That means factoring, expanding, combining like terms, applying exponent rules, and simplifying rational expressions. There is no equation to solve and no x to find. The answer is a different way of writing the same thing.

Advanced Math is where high scorers pull away from everyone else, and this skill is the foundation of the whole domain. If your algebra manipulation is shaky, you will also miss quadratics, systems, and function questions that depend on it. The College Board tests it directly because rewriting expressions cleanly is the single most reused move in SAT math.

The good news: these questions are extremely learnable. Almost every miss traces back to a small set of rule errors, like distributing an exponent over a sum or canceling terms that cannot be canceled. Fix those specific habits and this becomes one of the most reliable point sources on the test.

New to the Digital SAT? Start with how the test is structured and scored.

What the SAT actually tests

Expanding and factoring polynomials, including difference of squares like x² - 9 = (x + 3)(x - 3) and factoring quadratics into two binomials
Combining like terms and distributing correctly across parentheses, including a negative sign in front of a group
Exponent rules: product rule (add exponents), power rule (multiply exponents), quotient rule (subtract exponents), and negative or zero exponents
Rewriting radicals and fractional exponents, such as √x = x^(1/2) and x^(2/3) as the cube root of x²
Simplifying rational expressions by factoring the numerator and denominator, then canceling shared factors
Factoring with polynomial identities, like the sum and difference of cubes: a³ + b³ = (a + b)(a² - ab + b²), and dividing polynomials to rewrite a quotient as a whole part plus a remainder over the divisor
Choosing the equivalent form that reveals a specific feature, like a factored form that shows the zeros of an expression

Strategies that work

Plug in a test number

Pick an easy value like x = 2, evaluate the original expression, then evaluate each answer choice. Only the correct choice matches for every value you try, so if two choices match, test a second number. This turns an algebra question into arithmetic and catches sign errors you would never spot by staring.

Factor first, cancel second

With rational expressions, never cancel anything until both the numerator and denominator are fully factored. You can only cancel whole factors that multiply the entire top and bottom, never individual terms connected by plus or minus. Writing out the factored form takes ten seconds and prevents the most common trap in this skill.

Say the exponent rule out loud

Before touching a problem with powers, name the rule you are using: power to a power means multiply exponents, product of same bases means add them. Misapplied exponent rules cause more misses here than hard algebra does. Slowing down for one beat on the rule is faster than redoing the problem.

Work from the answers

If expanding the original looks messy, expand the answer choices instead and see which one rebuilds the original. Multiplying out a factored answer choice is usually mechanical and fast. This is especially useful on factoring questions where the choices differ only by signs.

Mistakes to avoid

Distributing an exponent over a sum: (x + y)² is not x² + y². It expands to x² + 2xy + y², and the missing middle term is the most common wrong answer on the page.
Canceling terms instead of factors in a fraction, like crossing out the x² in (x² - 9)/(x² + 5x + 6). Only shared factors cancel, and you can only see factors after factoring.
Multiplying a coefficient by the exponent instead of raising it to the power: (2x³)³ has a coefficient of 2³ = 8, not 2 times 3 = 6.
Dropping a negative when distributing, especially with a minus sign in front of parentheses: -(x - 5) becomes -x + 5, not -x - 5.
Adding exponents when the power rule calls for multiplying them: (x³)³ is x⁹, not x⁶.

Solve it with Desmos

Let the calculator do the work

The Digital SAT's built-in Desmos calculator can crack Equivalent Expressions questions in a few clicks. Here is the move that applies, step by step.

Check if Two Expressions Are EquivalentGraph both, and if they land on the exact same curve, they are equal.

All 15 Desmos techniques

Practice with Nova, your Math coach

Answer right here and Nova walks you through it, just like in a drill. Difficulty labels are relative within this skill.

Tip: the Calculator button on each question opens the same Desmos graphing calculator you get on the real Digital SAT. Try solving these with it.

Question 1Easy

The expression 3(x + 4) + 2x is given.

Which of the following is equivalent to the expression?

Pick an answer to get instant coaching from Nova.

Question 2Medium

The expression (2x³y²)³ / (4x²y) is given, where x and y are positive.

Which of the following is equivalent to the expression?

Pick an answer to get instant coaching from Nova.

Question 3Hard

The expression (x² - 9) / (x² + 5x + 6) is given, where x ≠ -2 and x ≠ -3.

Which of the following is equivalent to the expression?

Pick an answer to get instant coaching from Nova.

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The rest of Advanced Math

Equivalent Expressions is one of 3 skills in Advanced Math. The ideas build on each other, so it pays to learn them as a set rather than in isolation. See how Advanced Math fits together.

Other Math skills to explore

SAT is a registered trademark of the College Board, which was not involved in producing, and does not endorse, this product. Score estimates are Learner Labs projections, not official College Board scores.