Exponent Calculator

An exponent is a mathematical notation that represents the number of times a base number is multiplied by itself. It’s written as a small number to the upper right of the base number. For example, in 2³, 2 is the base and 3 is the exponent, meaning 2 × 2 × 2 = 8.

A negative exponent represents the reciprocal of the base raised to the positive exponent. For example, 2⁻³ = 1/(2³) = 1/8 = 0.125. The formula is: a⁻ⁿ = 1/(aⁿ).

Any non-zero number raised to the power of zero equals 1. This is known as the zero exponent rule. For example, 5⁰ = 1, 100⁰ = 1, and (-3)⁰ = 1. The only exception is 0⁰, which is undefined.

Fractional exponents represent roots. A number raised to the power of 1/n equals the nth root of that number. For example, 4^(1/2) = √4 = 2. For general fractions, a^(m/n) = (n√a)ᵐ = n√(aᵐ).

The main laws of exponents include: 1) Product Rule: aᵐ × aⁿ = a^(m+n), 2) Quotient Rule: aᵐ ÷ aⁿ = a^(m-n), 3) Power Rule: (aᵐ)ⁿ = a^(m×n), 4) Power of Product: (ab)ⁿ = aⁿ × bⁿ, 5) Power of Quotient: (a/b)ⁿ = aⁿ/bⁿ, and 6) Zero Exponent: a⁰ = 1 (for a ≠ 0).