# Indices

## The Laws of Indices

$x^a\times x^b=x^{a+b}$
$x^a\div x^b=x^{a-b}$
$\left(x^a\right)^b=x^{ab}$

$x^{-n}=\frac{1}{x^n}$

$x^{\frac{m}{n}}=\left(\sqrt[n]{x}\right)^m$

Here are some examples demonstrating how to use the above rules:

Example 1: $2^7\times 2^9=2^{16}$

Example 2: $\frac{4x^7}{2x^3}=2x^4$

Example 3: $\left(3p^2\right)^4=3^4(p^3)^3=81p^9$

Example 4: $16\times 2^{-3}=16\times \frac{1}{2^3}=16\times \frac{1}{8}=2$

Example 5: $\left(\frac{8}{27}\right)^{\frac{2}{3}}=\left(\left(\frac{8}{27}\right)^{\frac{1}{3}}\right)^2=\left(\sqrt[3]{\frac{8}{27}}\right)^2=\left(\frac{2}{3}\right)^2=\frac{4}{9}$