Equivalent expressions

Digital SAT · Math · Advanced Math. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

Two expressions are equivalent if they give the same value for every allowed value of the variable. These questions ask you to rewrite an expression in another form: expand, factor, combine fractions or use exponent laws.

For polynomials, multiply each term of one factor by each term of the other, then collect like terms. To factor, look first for a common factor, then for a difference of squares or a quadratic that splits into two factors.

For rational expressions, factor the top and the bottom and cancel only whole common factors. For powers and roots, write everything with the same base and use the laws of exponents.

Rules to remember

(a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b²
Difference of squares: a² − b² = (a − b)(a + b)
x^a × x^b = x^{a + b}, x^a ÷ x^b = x^{a − b}, (x^a)^b = x^ab
x^−a = 1/x^a and x⁰ = 1 (x ≠ 0)
x^1/n is the n-th root of x; x^m/n is the n-th root of x^m, so 8^2/3 = 4.
Adding fractions: a/b + c/d = (ad + bc)/(bd)
Quick check: put x = 2 into the original and into your answer; the values must match.

Common mistake

Cancelling terms instead of factors, for example crossing out x² in (x² − 9)/(x² + 5x + 6). Factor first, then cancel only a whole factor that appears on the top and the bottom.

Worked example

Which expression is equivalent to (x² − 9)/(x² + 5x + 6) for x ≠ −2 and x ≠ −3?

Factor the top: x² − 9 = (x − 3)(x + 3).
Factor the bottom: x² + 5x + 6 = (x + 2)(x + 3).
Cancel the common factor (x + 3): (x − 3)/(x + 2).
Check with x = 1: original = (1 − 9)/(1 + 5 + 6) = −8/12 = −2/3; answer = (1 − 3)/(1 + 2) = −2/3.

Answer: (x − 3)/(x + 2)

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