Laws of Indices

These are pretty important, especially for the rest of learning algebra.

An index (plural indices) is another word for saying numbers to the power of. We have already covered squared numbers $a^2$. Another word for them is exponent.

$a * a * a... = a^m$. This means that $a^m$ is the same saying $a * a * a$, $m$ number of times. For example $a^6 = a * a * a * a * a$

However there are various confusing rules that you need to learn. Where possible I will try and explain by using maths than by words. Remember that the letter being used doesn't mean anything, it just represents any possible number.

Basic Rules

$0^a = 0$ Because: $\color{red}{0 * 0 * 0 * 0... = 0}$

$1^b = 1$ Because: $\color{red}{1 * 1 * 1 * 1... = 1}$

$x^1 = x$

$a^0 = 1, a \neq 0$ Why? Well that's beyond GCSE.

Multiplying, Adding and Subtracting Rules

$x^mx^n = x^{m+n}$ Because: $\color{red}{x^4 x^5 = x^4 * x^5 =} \color{blue}{x * x * x * x} \color{black}{* x * x * x * x * x}$

$(x^m)^{n} = x^{mn}$ Example: $\color{red}{(x^4)^5 = x^{4*5} = x^{20}}$

Be careful on the last rule if you have other numbers like this example: $(5x^5)^{2} = 5\color{red}{^2}x\color{red}{^{5*2}} = 25x^{10}$ Remember to apply anything outside the brackets to everything in the brackets!

$a^m \div a^n = a^{m-n}$ Example: $\color{red}{\dfrac{a^5}{a^3} = a^{5-3} = a^2}$

Negative Exponent rules

$a^{-1} = \dfrac{1}{a}$ Example: $\color{red}{5^{-1} = \dfrac{1}{5^1} = {0.2}}$

$a^{-m} = \dfrac{1}{a^m}$ Because: $\color{red}{3^{-3} = 1 \div 3 \div 3 \div 3 = \dfrac{1}{3} \div \dfrac{3}{1} \div 3 = \dfrac{1}{3} * \dfrac{1}{3} \div 3 = \dfrac{1}{9} * \dfrac{1}{3} = \dfrac{1}{27}}$

Fraction Exponent rules

$a^{\tfrac{1}{n}} = \sqrt[n]{a}$ Example: $\color{red}{9^{\tfrac{1}{2}} = \sqrt{9^{1}} = 3}$

$a^{\tfrac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$

Applying the rules

You will need to remember all the stuff you have already learnt and combine it with these rules. $\begin{aligned} 18(f^{-4}) \div 9f^{-16} &= (\dfrac{18}{1} \times \dfrac{1}{f^4}) \div 9f^{-16} \\[7mu] \dfrac{18}{f^4} \div (\dfrac{9}{1} \times \dfrac{1}{f^{16}}) &= \dfrac{18}{f^4} \times \dfrac{f^{16}}{9} \\[7mu] \dfrac{18 \color{red}{(\div 3)}f^{16 \color{red}{-4}}}{9\color{red}{(\div 3)}f^{4 \color{red}{-4}}} &= \dfrac{6 \color{red}{(\div 3)}f^{12}}{3 \color{red}{(\div 3)}} = 2f^{12} \end{aligned}$