Exponent Calculator
Calculate any base raised to any power, including negative and fractional exponents.
The Formula
Rules:
aᵐ × aⁿ = aᵐ⁺ⁿ
(aᵐ)ⁿ = aᵐⁿ
a⁻ⁿ = 1/aⁿ
a^(1/n) = ⁿ√a
3⁻² = 1/9 ≈ 0.111
16^(1/4) = ⁴√16 = 2
(2³)² = 2⁶ = 64
Understanding Exponents
An exponent tells you how many times to multiply the base by itself. 2⁵ means 2 × 2 × 2 × 2 × 2 = 32. The base is 2, the exponent is 5, and the result is called the power. Exponents provide a compact way to express very large and very small numbers — 10⁹ (one billion) and 10⁻⁹ (one billionth) are far more readable than writing out all the digits.
Key Exponent Rules
Product rule: when multiplying same-base numbers, add exponents — aᵐ × aⁿ = aᵐ⁺ⁿ. Quotient rule: when dividing, subtract exponents — aᵐ ÷ aⁿ = aᵐ⁻ⁿ. Power rule: when raising a power to a power, multiply exponents — (aᵐ)ⁿ = aᵐⁿ. Zero exponent: any non-zero base to the power of 0 equals 1. Negative exponent: a⁻ⁿ = 1/aⁿ. Fractional exponent: a^(1/n) = the nth root of a.
Fractional and Negative Exponents
Fractional exponents express roots: 8^(1/3) = the cube root of 8 = 2. More generally, a^(m/n) = (nth root of a)^m. Negative exponents mean reciprocal: 4⁻² = 1/4² = 1/16 = 0.0625. These rules allow any root or reciprocal to be expressed in exponent notation, which is essential for calculus and algebra.
Frequently Asked Questions
What does a negative exponent mean?
A negative exponent means the reciprocal: 2u207bu00b3 = 1/2u00b3 = 1/8 = 0.125. It does not make the result negative.
What is anything to the power of 0?
Any nonzero number raised to the power of 0 equals 1. This is because dividing a number by itself (au207f/au207f = au2070) always equals 1.