Introduction
Exponent and radical manipulation appears throughout the Advanced Math domain, which makes up roughly 35% of your SAT Math score. Every time the test gives you an expression and asks "which of the following is equivalent to…," it is probing your ability to fluently apply exponent laws and convert between radical and rational-exponent forms. These skills also underpin polynomial work, factoring, and rational expressions — the sibling subtopics covered in their own notes (Polynomial Operations, Factoring Polynomials, Rational Expressions).
Core Concept
The central idea is that exponents, radicals, and rational exponents are three notations for the same underlying idea: repeated multiplication and its inverse. Switching fluently among them — without a calculator — is what the SAT tests.
The Two-Way Bridge: Rational Exponents ↔ Radicals
The denominator of the rational exponent is the index (root), and the numerator is the power. For example:
You can also compute it the other way: — same answer. Choose whichever order keeps the numbers smaller.
Simplifying Radical Expressions
Factor the radicand to pull out perfect powers:
To combine radical terms, first simplify each one, then collect like-radical terms exactly as you would collect like terms in algebra.
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