Fractional powers

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 $b$ .