Laws of Exponents — SAT Math Explained
A set of rules governing how to simplify expressions involving powers (exponents) when multiplying, dividing, raising to powers, or using negative/zero exponents.
The Core Idea
Exponent rules aren't arbitrary — each one follows logically from the definition of exponentiation as repeated multiplication. Understanding why each rule works prevents memorization errors.
Laws
aᵐ × aⁿ = aᵐ⁺ⁿ — same base: add exponents
aᵐ ÷ aⁿ = aᵐ⁻ⁿ — same base: subtract exponents
(aᵐ)ⁿ = aᵐⁿ — power to a power: multiply exponents
(ab)ⁿ = aⁿbⁿ — distribute the exponent to each factor
(a/b)ⁿ = aⁿ/bⁿ — distribute the exponent to numerator and denominator
a⁰ = 1 (for a ≠ 0) — anything to the zero power is 1
a⁻ⁿ = 1/aⁿ — negative exponents mean reciprocals, NOT negatives
Why Zero Exponent Is One
aⁿ/aⁿ = aⁿ⁻ⁿ = a⁰, and any number divided by itself = 1, so a⁰ = 1
Why Negative Exponents Are Reciprocals
a¹/a³ = a¹⁻³ = a⁻², but also = 1/a², so a⁻² = 1/a²
Real World Application
Scientific notation, compound interest, population growth modeling, computing in programming
Common Errors to Avoid
Adding exponents when bases are different (rule only applies to same base)
Thinking a⁻² = -a² instead of 1/a²
Distributing an exponent to an addition inside parentheses: (a + b)² ≠ a² + b²
Practice: Laws of Exponents
5 SAT-style questions. Select your answer and get an instant explanation.
Simplify: x³ · x⁴
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