Exponent Calculator

Calculate any base raised to any power. Supports integers, decimals, negative exponents, and fractional exponents with step-by-step explanation.

How It Works

Exponentiation is the operation of raising a base b to a power n, written as bⁿ. It represents repeated multiplication of the base:

bⁿ = b × b × b × ... × b   (n times)

For example, 2⁵ = 2 × 2 × 2 × 2 × 2 = 32.

Special Cases

b⁰ = 1 — Any nonzero number raised to the power of 0 equals 1
b¹ = b — Any number raised to the power of 1 equals itself
b⁻ⁿ = 1 / bⁿ — A negative exponent means the reciprocal
b^1/n = ⁿ√b — A fractional exponent means taking a root

Exponent Rules

b^m × bⁿ = b^m+n (Product Rule)

b^m / bⁿ = b^m−n (Quotient Rule)

(b^m)ⁿ = b^m×n (Power Rule)

(a × b)ⁿ = aⁿ × bⁿ (Product of Powers)

(a / b)ⁿ = aⁿ / bⁿ (Quotient of Powers)

b⁰ = 1 (Zero Exponent, b ≠ 0)

b⁻ⁿ = 1 / bⁿ (Negative Exponent)

Common Powers

b	b²	b³	b⁴	b⁵	b¹⁰
2	4	8	16	32	1,024
3	9	27	81	243	59,049
4	16	64	256	1,024	1,048,576
5	25	125	625	3,125	9,765,625
10	100	1,000	10,000	100,000	10,000,000,000

Applications of Exponents

Compound Interest — A = P(1 + r)^t calculates investment growth over time
Computer Science — Powers of 2 define memory sizes (1 KB = 2¹⁰ bytes)
Physics — Inverse-square law: intensity decreases as 1/r²
Population Growth — Exponential growth models use P × e^rt
Scientific Notation — Large numbers expressed as a × 10ⁿ

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