Negative Exponents

Negative exponents mean that instead of multiplying that many of the base number together, you divide.

For example, ${5}^{2}=25$ but ${5}^{-2}=\frac{1}{{5}^{2}}=\frac{1}{25}$ .

You can use the product of powers property to show this.

${5}^{-2}×{5}^{2}={5}^{\left(-2 + 2\right)}={5}^{0}$

We know ${5}^{2}=25$ , and we know ${5}^{0}=1$ . So, this says that ${5}^{-2}×25=1$ . What number times $25$ equals $1$ ? That would be its multiplicative inverse , $\frac{1}{25}$ .

${5}^{-2}=\frac{1}{25}$

In general, for all real numbers $a$ and $b$ , where $a\ne 0$ , ${a}^{\mathit{- b}}$ is defined to be $\frac{1}{{a}^{b}}$ , so

${a}^{\mathit{- b}}=\frac{1}{{a}^{b}}$