Surds — simplifying and rationalising

Key learnings

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  Simplify by pulling out the largest square factor

  $\sqrt{ab} = \sqrt{a}\,\sqrt{b}$, so $\sqrt{72} = \sqrt{36}\,\sqrt{2} = 6\sqrt{2}$. Take out the biggest perfect square in one go.

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  Only like surds add

  $a\sqrt{c} + b\sqrt{c} = (a+b)\sqrt{c}$, but $\sqrt{a} + \sqrt{b} \ne \sqrt{a+b}$. Simplify each surd first to spot like terms.

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  Rationalise to clear a surd from the denominator

  For $\dfrac{1}{\sqrt{b}}$ multiply top and bottom by $\sqrt{b}$; for $\dfrac{1}{p+\sqrt{q}}$ multiply by the conjugate $p-\sqrt{q}$.