Exponents, Powers, Roots: A Practical Refresher

The core rules

• a^m · a^n = a^(m+n)
• (a^m)^n = a^(m·n)
• a^(−n) = 1 / a^n
• a^0 = 1 (for a ≠ 0)
• a^(1/n) = nth root of a

Why a^0 = 1

Because a^n / a^n = 1 and also equals a^(n−n) = a^0. The definition preserves the pattern.

Practical uses

Compound interest, exponential growth, decibels, pH, earthquake magnitudes — all logarithmic or exponential. Understanding the rules once unlocks every domain.

The scientific-calculator button map

^ or x^y raises to a power. √ takes square root. y√x on a scientific calculator takes any root. Our exponent and square-root calculators handle both.