Exponent Rules

Definition For a real number $a$ and a positive integer $n$, the power $a^n$ is the product of $n$ factors equal to $a$,

The number $a$ is the base and $n$ is the exponent. The exponent applies only to the symbol directly below it, so $-3^2 = -(3^2) = -9$, while $(-3)^2 = 9$, and $2x^3$ means $2 \cdot x^3$, not $(2x)^3$.

Definition For a real number $a$ and positive integers $m$ and $n$,

The rule also holds for all integers $m$ and $n$ when $a \ne 0$, using the definitions of zero and negative exponents below. It needs the same base on both factors.

Definition For a real number $a \ne 0$ and integers $m$ and $n$ (zero and negative exponents are defined below),

An expression simplified this way keeps the restriction of the original. For example $\frac{x^5}{x^2}$ is undefined at $x = 0$, so it equals $x^3$ only for $x \ne 0$.

Definition For a real number $a$ and positive integers $m$ and $n$,

For real numbers $a$ and $b$ and a positive integer $n$,

These also hold for all integers when the bases involved are nonzero. They apply to products and quotients only. A power of a sum is not the sum of the powers, so in general $(a + b)^2 \ne a^2 + b^2$.

Definition For every real number $a \ne 0$,

This is the only choice that keeps the quotient rule true, since $\frac{a^n}{a^n} = 1$ and also $\frac{a^n}{a^n} = a^{n - n} = a^0$. The expression $0^0$ is left undefined here, so a statement like $x^0 = 1$ carries the condition $x \ne 0$.

Definition For a real number $a \ne 0$ and a positive integer $n$,

A negative exponent never makes a number negative. It stands for a reciprocal. In particular $a^{-1} = \frac{1}{a}$, and for $a \ne 0$ and $b \ne 0$, $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$. As with sums of powers, $(a + b)^{-1}$ is not $a^{-1} + b^{-1}$.