# Indices

Indices are also known as powers or orders. Note that indices is plural and index is singular. It can be shown that indices abide by the following rules in maths:

## The Laws of Indices

• $x^a\times x^b=x^{a+b}$
• $x^a\div x^b=x^{a-b}$
• $\left(x^a\right)^b=x^{ab}$
• $x^{-n}=\frac{1}{x^n}$
• $x^{\frac{m}{n}}=\left(\sqrt[n]{x}\right)^m$

### Example 1

• $2^7\times 2^9=2^{16}$
• $\frac{4x^7}{2x^3}=2x^4$
• $\left(3p^2\right)^4=3^4(p^2)^4=81p^8$
• $16\times 2^{-3}=16\times \frac{1}{2^3}=16\times \frac{1}{8}=2$

### Example 2

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