The product of two terms with exponents and the same base

is equal to the base raised to the sum of the exponents.

$\begin{array}{rcl}{x}^{m}{x}^{n}& =& {x}^{\text{(}m+n\text{)}}\\ {2}^{3}{2}^{4}& =& {2}^{\text{(}3+4\text{)}}\\ & =& {2}^{7}\end{array}$

The quotient of two terms with exponents and the same base

is equal to the base raised to the difference of the exponents.

$\begin{array}{rcl}{x}^{m}\xf7{x}^{n}& =& {x}^{\text{(}m-n\text{)}}\\ {2}^{4}\xf7{2}^{3}& =& {2}^{\text{(}4-3\text{)}}\\ & =& {2}^{1}\end{array}$

A term with an exponent raised to a power is equal

to the base raised to the product of the exponents.

$\begin{array}{rcl}{\text{(}{x}^{m}\text{)}}^{n}& =& {x}^{\text{(}m\times n\text{)}}\\ {\text{(}{2}^{3}\text{)}}^{4}& =& {2}^{\text{(}3\times 4\text{)}}\\ & =& {2}^{12}\end{array}$

A term with a negative exponent is equal to the reciprocal

of the base raised to the opposite of the exponent.

$\begin{array}{rcl}{x}^{-m}& =& \frac{1}{{x}^{m}}\\ {2}^{-3}& =& \frac{1}{{2}^{3}}\end{array}$