Indices and surds

The idea

Index laws work for any powers: $a^m \times a^n = a^{m+n}$, $a^{-n} = \frac{1}{a^n}$ and $a^{\frac{m}{n}} = (\sqrt[n]{a})^m$. Surds stay exact: simplify $\sqrt{ab} = \sqrt{a}\sqrt{b}$ and rationalise denominators.

1. Write every number as a power of the same base when you can: $8 = 2^3$.
2. Take square factors out of a surd: $\sqrt{50} = 5\sqrt{2}$.
3. Rationalise $\frac{1}{a + \sqrt{b}}$ by multiplying top and bottom by $a - \sqrt{b}$.

Watch out Keep answers exact until the end: a rounded decimal loses the marks for 'exact'.