PSAT 8/9 Math: Equivalent Expressions
Writing equivalent expressions means rewriting an expression in a different but mathematically identical form, and the PSAT 8/9 tests this through combining terms, distributing, factoring, and applying exponent rules. This free, complete lesson covers combining like terms, the distributive property, factoring out a greatest common factor, and exponent rules and rational exponents. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback and full explanations. Everything here is free, and you can keep practicing afterward with the linked quiz below.
1. Combining Like Terms
Like terms have the exact same variable(s) raised to the exact same power(s). Combine them by adding or subtracting their coefficients, while keeping the variable part unchanged.
Worked Example: Simplify 8x + 5y + 2x + 7y.
Group like terms: (8x + 2x) + (5y + 7y).
= 10x + 12y
1. Simplify 8x + 5y + 2x + 7y. (see worked example above)
Explanation: Combine the x-terms: 8x + 2x = 10x. Combine the y-terms: 5y + 7y = 12y. Result: 10x + 12y.
2. Simplify 14x³ − 9x³.
Explanation: These are like terms (same variable, same exponent), so subtract coefficients: 14 − 9 = 5, keeping x³: 5x³.
3. Add (x + 7) and (3x − 5), then simplify.
Explanation: Combine x-terms: x + 3x = 4x. Combine constants: 7 − 5 = 2. Result: 4x + 2.
2. The Distributive Property
The distributive property says a(b + c) = ab + ac. Multiply the term outside the parentheses by every term inside, being careful with signs, especially when distributing a subtraction.
Worked Example: Expand 12(x² − 5).
Distribute 12 to each term: 12x² − 60
1. Expand 12(x² − 5). (see worked example above)
Explanation: Distribute 12 across both terms: 12 × x² = 12x², and 12 × (−5) = −60. Result: 12x² − 60.
2. Simplify 18w − (6w + 4w).
Explanation: First combine inside the parentheses: 6w + 4w = 10w. Then 18w − 10w = 8w.
3. Expand 4(x + 6), then subtract 8.
Explanation: Distribute: 4(x + 6) = 4x + 24. Then subtract 8: 4x + 24 − 8 = 4x + 16.
3. Factoring Out a Greatest Common Factor
To factor an expression, find the greatest common factor (GCF) shared by every term, then write the expression as that GCF multiplied by what's left over in parentheses.
Worked Example: Factor 6x² + 9x.
The GCF of 6x² and 9x is 3x.
6x² + 9x = 3x(2x + 3)
1. Factor 6x² + 9x. (see worked example above)
Explanation: The GCF of 6x² and 9x is 3x. Dividing each term by 3x: 6x²/3x = 2x, and 9x/3x = 3. Result: 3x(2x + 3).
2. Factor x³ + 6.
Explanation: x³ and 6 share no common variable or numerical factor besides 1, so this binomial cannot be factored using a simple GCF.
3. Factor 15x²y + 10xy.
Explanation: The GCF of 15x²y and 10xy is 5xy. Dividing each term: 15x²y/5xy = 3x, and 10xy/5xy = 2. Result: 5xy(3x + 2).
4. Exponent Rules and Rational Exponents
When multiplying powers with the same base, add the exponents: am × an = am+n. A rational exponent like x1/n is equivalent to an nth root: x1/n = n√x.
Worked Example: Rewrite x1/5 (for x > 0) as an equivalent radical expression.
x1/5 = 5√x (the fifth root of x)
1. Rewrite x1/5 (for x > 0) as an equivalent radical expression. (see worked example above)
Explanation: A rational exponent of 1/n corresponds to an nth root, so x1/5 = 5√x.
2. Multiply the monomials: (6x)(2y)(3y), simplifying the result.
Explanation: Multiply coefficients: 6 × 2 × 3 = 36. Multiply variables: x stays as x, and y × y = y² (adding exponents of the repeated base). Result: 36xy².
3. Simplify x&sup4; × x³.
Explanation: When multiplying powers with the same base, add the exponents: x&sup4; × x³ = x⁴⁺³ = x⁷.
Common Mistakes to Avoid
- Combining terms that aren't actually "like terms," such as adding x and x² together, their exponents must match, not just their base variable.
- Forgetting to distribute a negative sign to every term inside parentheses, especially in expressions like 18w − (6w + 4w), where the subtraction applies to both terms inside.
- Factoring out a GCF that isn't actually the greatest common factor, always check that each term's GCF includes every shared variable and the largest shared number.
- Multiplying exponents instead of adding them (or vice versa) when combining powers with the same base, remember: am × an = am+n, not amn.
- Misreading a rational exponent's denominator as the root's index. x1/n means the nth root of x, so x1/3 is a cube root, not a third power.
Frequently Asked Questions
How do I know when two terms are "like terms"?
Like terms must have the exact same variable(s) raised to the exact same exponent(s). For example, 3x² and 7x² are like terms, but 3x² and 7x are not, since their exponents differ.
What's the difference between factoring and distributing?
Distributing starts with a factored form (like 3x(2x + 3)) and expands it into separate terms (6x² + 9x). Factoring does the reverse: it starts with separate terms and rewrites them as a product, by pulling out a common factor.
Where can I practice more problems like these?
The Equivalent Expressions quizzes in the PSAT 8/9 Math Question Bank include additional original problems on this topic, along with quizzes covering every other Heart of Algebra, Advanced Math, Problem-Solving and Data Analysis, and Geometry and Trigonometry skill tested on the PSAT 8/9. You can also browse the full PSAT 8/9 Math Question Bank.