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SAT Math Unit 3: Equivalent expressions

Equivalent-expression questions test algebra fluency: expanding, factoring and simplifying without errors.

Covers: Exponent rules, polynomial operations, factoring, rational expressions, matching coefficients

Study guide

Equivalent expressions are different-looking expressions that are equal for every value of the variable. The exponent rules are the core tools: multiply powers by adding exponents (xᵃ·xᵇ = xᵃ⁺ᵇ), divide by subtracting (xᵃ/xᵇ = xᵃ⁻ᵇ), and raise a power to a power by multiplying ((xᵃ)ᵇ = xᵃᵇ). A negative exponent means a reciprocal, x⁰ = 1, and a fractional exponent means a root: x^(1/2) = √x.

Polynomial operations come down to combining like terms and distributing. Know the special products: (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b², and (a + b)(a − b) = a² − b². Factoring reverses these: pull out the greatest common factor first, then look for a difference of squares or a trinomial that factors into two binomials.

Rational expressions are fractions with polynomials. Simplify by factoring the top and bottom and canceling common factors, but remember any value that made the original denominator zero is still excluded. Finally, if two polynomial expressions are equal for all x, their matching coefficients are equal. That lets you find unknown constants by expanding one side and comparing terms.

In general, x^(m/n) = ⁿ√(xᵐ) = (ⁿ√x)ᵐ, so 8^(2/3) = (∛8)² = 4. To simplify radicals, use √(ab) = √a·√b and pull out perfect-square factors, as in √50 = 5√2. A fraction like (x² + 5x + 7)/(x + 2) can be rewritten by long division as x + 3 + 1/(x + 2). The SAT asks for this form when answer choices show a polynomial plus a remainder over the divisor.

To factor ax² + bx + c when a ≠ 1, find two numbers that multiply to ac and add to b, split the middle term with them, and factor by grouping. Always check by multiplying back out.

Key ideas

• Combine like terms only when variables and exponents match exactly.
• FOIL or distribute carefully; check the middle term.
• If two polynomials are equal for all x, their matching coefficients are equal.
• Factor differences of squares and trinomials to simplify fractions.
• Exponent rules: add when multiplying, subtract when dividing, multiply when raising a power.

Formulas and rules

Common mistakes

• Dropping the middle term when squaring a binomial.
• Canceling terms instead of factors in a fraction.

Words to know

Square of a sum

(a + b)² = a² + 2ab + b²

Matching coefficients

If two polynomials are equal for all x, their like-term coefficients are equal.

Quotient of powers

xᵃ / xᵇ = xᵃ⁻ᵇ

Difference of squares

a² − b² = (a − b)(a + b)

Power of a power

(xᵃ)ᵇ = xᵃᵇ

Equivalent expressions

Expressions equal for every value of the variable.

Zero exponent

x⁰ = 1 for any x ≠ 0.

Fractional exponent

x^(1/n) = ⁿ√x, so x^(1/2) = √x.

Square of a difference

(a − b)² = a² − 2ab + b²

Rational expression

A fraction with polynomials; values that make the denominator 0 are excluded.

Like terms

Terms with the same variable and exponent, which can be combined.

Factoring strategy

Pull out the GCF first, then look for a difference of squares or a factorable trinomial.

Sample practice questions

141 more questions like these, from level 1 to 5, are in the practice set.