### Calculating with numbers: Calculating with powers and roots

### Fractional powers

Calculation rules for powers with integers as exponents have been discussed before. These can be extended to rational exponents in case of a positive base.

For any fraction \(\tfrac{m}{n}\) with \(n>1\) we define the **fractional power** \[a^{\frac{m}{n}}=\sqrt[n]{a^m}\]

**Examples**

\[\begin{aligned}2^{\frac{4}{3}}&=\sqrt[3]{2^4}=2\sqrt[3]{2}\\[0.2cm] 2^{-\frac{1}{2}}&=\sqrt{2^{-1}}=\sqrt{\frac{1}{2}}=\frac{1}{2}\sqrt{2}\end{aligned}\]

Calculation rules for fractional powers \[\begin{aligned}a^r \times a^s &= a^{r+s} \\ \\

\frac{a^r}{a^s}&= a^{r-s} \\ \\

\left(a^r\right)^s&= a^{r\times s} \\ \\

(a \times b)^r&= a^r \times b^r \\ \\

\left(\frac{a}{b}\right)^r&= \frac{a^r}{b^r}\end{aligned}\] for all rational numbers \(r\) and \(s\) and all positive numbers \(a\) and \(b\) .