Surds

The idea

A surd is a root that cannot be written as a whole number or a fraction (an irrational root), left exact, like $\sqrt{2}$. Simplify by taking out square factors: $\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$. Rationalise $\frac{a}{\sqrt{b}}$ by multiplying top and bottom by $\sqrt{b}$.

1. $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$.
2. Find the largest square factor to simplify.
3. $\frac{6}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}$; for $\frac{1}{2 + \sqrt{3}}$, multiply by $2 - \sqrt{3}$.

Watch out $\sqrt{9 + 16}$ is $5$, not $3 + 4$: roots do not split over addition.