Laws of indices
Introduction
You need to be familiar with these laws of indices.
\(x^a × x^b = x^{a + b}\).
\(x^a ÷ x^b = x^{a - b}\).
\((x^a)^b = x^{a × b}\).
Learn these special cases off by heart.
Anything to the power of 1 is itself, i.e. \(x^{1} = x\).
Anything to the power of 0 is 1, i.e. \(x^{0} = 1\).
\(x^{-1} = \frac{1}{x}\).
How to apply the laws of indices
Example 1 - Multiplying powers
a) Simplify \(5^6 × 5^7\).
When multiplying, add together the powers:
\begin{aligned} 5^6 × 5^7 & = 5^{6 + 7}\\ & = 5^{13}. \end{aligned}
b) Simplify \(3a^2 × 4a^7\).
Multiply the numbers as normal, then add together the powers:
\begin{aligned} 3x^2 × 4x^7 & = 12x^{2 + 7}\\ & = 12x^9. \end{aligned}
Example 2 - Dividing powers
Simplify \(x^8 ÷ x^2\).
When dividing, subtract the powers:
\begin{aligned} x^8 ÷ x^2 & = x^{8 - 2}\\ & = x^6. \end{aligned}
Example 3 - Raising a power to another power
Simplify \((x^2)^6\).
When raising a power to another power, multiply the powers:
\begin{aligned} (x^2)^6 & = x^{2 × 6}\\ & = x^{12}. \end{aligned}
Worksheets to practise applying the basic laws of indices
Try this worksheet to practise your skills.