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}.
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