Exponent rules
Also known as: laws of exponents, exponent properties
The exponent rules are the properties that let you simplify expressions containing powers — including multiplying like bases by adding exponents, dividing by subtracting them, and raising a power to a power by multiplying them. They apply to any nonzero base.
An exponent tells you how many times a base is multiplied by itself, so 2⁵ means 2 · 2 · 2 · 2 · 2 = 32. The exponent rules are shortcuts that follow directly from that definition, letting you combine and simplify powers without expanding them.
The core rules are: product rule, xᵃ · xᵇ = xᵃ⁺ᵇ (multiply like bases, add exponents); quotient rule, xᵃ / xᵇ = xᵃ⁻ᵇ (divide like bases, subtract exponents); power rule, (xᵃ)ᵇ = xᵃᵇ (raise a power to a power, multiply exponents); distributed power, (xy)ᵃ = xᵃyᵃ and (x/y)ᵃ = xᵃ/yᵃ; zero exponent, x⁰ = 1 for any nonzero x; negative exponent, x⁻ᵃ = 1/xᵃ; and fractional exponent, x^(a/b) = the b-th root of xᵃ, which is how radicals and exponents connect.
The single most common error is applying the product or quotient rule to unlike bases. 2³ · 5³ does not simplify by adding exponents — the bases differ, so use (xy)ᵃ = xᵃyᵃ in reverse to get 10³. A close second is mishandling signs: (−3)² = 9, but −3² = −9, because the exponent binds to the 3 before the negative sign is applied. Note also that xᵃ + xᵇ has no simplification rule; the properties govern multiplication and division, not addition.
Exponent rules are foundational on the GRE quantitative section, the SAT math sections, and CLT algebra, where they appear both directly and hidden inside exponential functions, radicals, and scientific notation problems. Because these tests reward speed, knowing the rules cold is worth more than deriving them under time pressure.
Key takeaways
- Multiply like bases by adding exponents; divide like bases by subtracting them.
- Raise a power to a power by multiplying the exponents, and distribute an exponent across a product or quotient.
- Any nonzero base to the zero power equals 1, and a negative exponent means the reciprocal.
- A fractional exponent is a root: x^(a/b) is the b-th root of xᵃ.
- The rules apply only to like bases and only to multiplication and division — never to sums.