Surds and Indices

Powers (indices) and roots (surds) are the language of algebra. Getting fluent here makes every later chapter easier, and these skills are tested directly in the non-calculator style "show all stages of working" questions.

The laws of indices

For any base $a$ and powers $m, n$:

$a^m \times a^n = a^{m+n}$ — add the powers when multiplying
$a^m \div a^n = a^{m-n}$

Worked example. Simplify $\dfrac{6x^5}{2x^2}$. Divide the numbers and subtract the powers: $3x^{5-2} = 3x^3$

Worked example. Evaluate $16^{3/4}$. The denominator is the root, the numerator is the power: $16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8$