Indices & Surds

Indices

$1.\,a^m \times a^n = a^{m+n}$
$2.\,a^m \div a^n = a^{m-n}$
$3.\,(a^m)^n = a^{mn}$
$4.\,a^0 = 1 \mbox{ where } a \neq 0$
$5.\,a^{-n} = \frac{1}{a^n}$
$6.\,a^{\frac{1}{n}} = \sqrt[n]{a}$
$7.\,a^{\frac{m}{n}} = (\sqrt[n]{a})^m$
$8.\,(a \times b)^n = a^n \times b^n$
$9.\,(\frac{a}{b})^n = \frac{a^n}{b^n}$

Surds

$1.\,\sqrt{a} \times {\sqrt{a}} = a$
$2.\,\sqrt{a} \times {\sqrt{b}} = \sqrt{ab}$
$3.\,\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$
$4.\,m \sqrt{a} + n \sqrt{a} = (m+n) \sqrt{a}$
$5.\,m \sqrt{a} - n \sqrt{a} = (m-n) \sqrt{a}$

Rationalise Denominator

For $\dfrac{k}{a\sqrt{b}}$ , multiply numerator and denominator by $\sqrt{b}$ .
For $\dfrac{k}{a\sqrt{b}+c\sqrt{d}}$ , multiply by the conjugate, which is $a\sqrt{b}-c\sqrt{d}$ .