Last updated: 2026-06-04

What is an exponent?

Exponentiation expresses repeated multiplication. The expression $b^n$ means the base $b$ is multiplied by itself $n$ times:

The number $b$ is the base and $n$ is the exponent (also called the power or index). For example, $2^{10} = 1024$ because 2 is multiplied by itself 10 times.

The fundamental exponent rules

These six rules govern how exponents behave under arithmetic operations. Each follows directly from the definition of repeated multiplication.

Product rule — add exponents when multiplying the same base: $b^m \cdot b^n = b^{m+n}$

Quotient rule — subtract exponents when dividing: $\frac{b^m}{b^n} = b^{m-n} \quad (b \neq 0)$

Power of a power — multiply exponents when exponentiating a power: $(b^m)^n = b^{mn}$

Power of a product — distribute the exponent over a product: $(ab)^n = a^n \cdot b^n$

Zero exponent — any non-zero base raised to 0 equals 1: $b^0 = 1 \quad (b \neq 0)$

Negative exponent — a negative exponent indicates a reciprocal: $b^{-n} = \frac{1}{b^n} \quad (b \neq 0)$

Negative exponents

A negative exponent means "take the reciprocal." For example:

$2^{-3} = \frac{1}{2^3} = \frac{1}{8} = 0.125$

This follows from the quotient rule: $b^0 / b^n = 1 / b^n = b^{-n}$. Negative exponents appear in scientific notation for very small numbers ($10^{-9}$ m = 1 nanometre) and in dimensional analysis ($\text{m}\,\text{s}^{-1}$ for speed).

Fractional exponents and roots

A fractional exponent represents a root. The general rule is:

$b^{m/n} = \sqrt[n]{b^m} = \left(\sqrt[n]{b}\right)^m$

Common cases:

For fractional exponents to produce a real number, the base must be non-negative (or the exponent must simplify to a ratio with an odd denominator for negative bases). Complex results fall outside the scope of standard arithmetic.

The reciprocal

The reciprocal of $b^n$ is $b^{-n} = 1/b^n$. It satisfies $b^n \cdot b^{-n} = b^0 = 1$. The reciprocal is undefined when $b^n = 0$ (that is, when $b = 0$ and $n > 0$).

0⁰ — the convention

When $b = 0$ and $n = 0$, the expression $0^0$ is an indeterminate form in analysis (because $0^x \to 0$ and $x^0 \to 1$ from different directions). In discrete mathematics, combinatorics, and programming, $0^0$ is universally defined as 1. This choice keeps combinatorial identities (binomial theorem, power series) consistent at boundary cases. This calculator follows the convention $0^0 = 1$.

Worked example

Inputs: base $b = 3$, exponent $n = 4$

$b^n = 3^4 = 3 \times 3 \times 3 \times 3 = 81$

Reciprocal:

$b^{-n} = 3^{-4} = \frac{1}{81} \approx 0.012346$

Verification using the product rule:

$3^4 \cdot 3^{-4} = 3^{4 + (-4)} = 3^0 = 1 \checkmark$

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