An exponent records repeated multiplication. For a positive integer $n$, the expression $a^n$ means multiplying $a$ by itself $n$ times. For example,

$$a^4=a\cdot a\cdot a\cdot a.$$

Product and quotient laws

Repeated multiplication explains why powers with the same base combine by adding or subtracting exponents:

$$a^m a^n=a^{m+n},$$

$$\frac{a^m}{a^n}=a^{m-n},\qquad a\ne0.$$

A power raised to another power multiplies the exponents:

$$(a^m)^n=a^{mn}.$$

Products can also be raised term by term:

$$(ab)^n=a^n b^n.$$

Zero and negative exponents

The same laws remain consistent when exponents are extended beyond positive integers. For $a\ne0$,

$$a^0=1$$

and

$$a^{-n}=\frac1{a^n}.$$

For example,

$$2^{-3}=\frac1{2^3}=\frac18.$$

Exponent laws are rules for rewriting powers without changing their value. The restrictions on denominators still apply: expressions such as $0^{-1}$ are undefined.