Exponents in Algebraic Contexts

GRE Quantitative · Axiom Academy

LESSON Exponents in Algebraic Contexts Every exponent rule you already know still applies once variables and equations enter the picture — the base can change; the rule never does. 1. The Rules Don't Change — Only the Base Does The product rule , the power rule (x^m)^n = x^ mn , and "distribute across a product" (xy)^n = x^n y^n are the same rules whether the base is a number or a variable. Watch the identical rule-path run on a numeric base and an algebraic base at once — the shape of the simplification never changes. Algebraic instance — same rule, same steps Simplify . Distribute the outer power across every factor, then apply the product rule to the leftover x 's: 2. Negative and Fractional Exponents Combine the Same Way and are just rewrite rules — once applied, the exponents underneath still add exactly like whole numbers. Watch the two signed exponents join tip-to-tail on the number line as the algebraic product collapses to a single power. Simplify . The bases match, so add the exponents: . 3. Solving Exponential Equations by Matching Bases If b^m = b^n for the same positive base , then m = n — the exponents themselves must be equal. So the strategy for a GRE exponential equation is: rewrite both sides with a common base , then set the exponents equal and solve the resulting (usually linear) equation. Watch the two towers lock into alignment only once the exponents match.

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