By Raleigh Kung | Updated Aug 30, 2022
Exponents—often seen as superscript numbers or symbols next to a base—represent repeated multiplication. While the concept is simple, mastering the rules that govern exponents is essential for success in algebra, calculus, and real‑world problem solving.
In exponentiation, the base is the ordinary number or variable, and the exponent (written in superscript) indicates how many times the base is multiplied by itself. For instance, the expression 5 × 5 × 5 is the expanded form of the exponentiation 53.
When solving expressions, exponents are addressed immediately after parentheses but before multiplication or division. Complex exponential expressions are treated as self‑contained equations and are simplified before the surrounding arithmetic.
Numbers raised to the power of 2 are called squared, while those raised to the power of 3 are cubed. Exponents of 1 leave the number unchanged, and any non‑zero number raised to the power of 0 equals 1.
To add or subtract like terms, the bases and exponents must match. For example, x2 + x2 equals 2x2. However, x2 + x3 cannot be combined directly; the terms must first be factored or rewritten so that they share a common base and exponent.
When multiplying terms with the same base, simply add their exponents: x2 × x2 = x4. For division, subtract the exponents: x3 ÷ x2 = x1 (or just x). A negative exponent indicates the reciprocal: x-2 = 1/x2.
Exponential functions underpin many scientific and financial models. The radioactive half‑life of a substance follows an exponential decay curve. In finance, compound interest and stock growth are modeled with exponential rates. Even everyday phenomena—such as the braking distance of a vehicle—exhibit exponential behavior: doubling speed more than doubles the stopping distance.
