Simplify surds, carry out the four operations with surds, expand brackets containing surds, and rationalise the denominator of a fraction (Higher tier).

What this dot point is asking

OCR places surds in the Higher-tier number content (an extension of N8). A surd is a root that cannot be written exactly as a fraction, so working with surds keeps answers exact rather than rounding. You must simplify surds, add, subtract, multiply and divide them, expand brackets that contain them, and rationalise denominators. Surds appear on the non-calculator paper and feed straight into the quadratic formula, Pythagoras and trigonometry with exact values, so they are a high-value Higher topic.

What a surd is

An irrational number cannot be written as an exact fraction; its decimal neither terminates nor recurs. The square root of any whole number that is not a perfect square is irrational, so $\sqrt{2}, \sqrt{3}, \sqrt{5}$ are surds, but $\sqrt{9} = 3$ is not. Leaving an answer as a surd is leaving it exact, which is why exam questions say "give your answer in surd form" or "in the form $a\sqrt{b}$".

Simplifying surds

The key move is to split out a square factor.

So $\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$. Choosing the largest square factor finishes in one step; using a smaller one ($\sqrt{72} = \sqrt{4 \times 18} = 2\sqrt{18}$) means you must simplify again.

Adding, subtracting and multiplying

Surds behave like algebraic terms: only like surds combine.

For addition and subtraction, simplify first so that matching surds appear, then add the coefficients: $\sqrt{12} + \sqrt{27} = 2\sqrt{3} + 3\sqrt{3} = 5\sqrt{3}$. For multiplication, multiply coefficients together and surds together: $2\sqrt{5} \times 3\sqrt{2} = 6\sqrt{10}$. When you multiply a surd by itself the root disappears: $\sqrt{7} \times \sqrt{7} = 7$. Expanding brackets follows the usual rules, for example $(2 + \sqrt{3})(1 + \sqrt{3}) = 2 + 2\sqrt{3} + \sqrt{3} + 3 = 5 + 3\sqrt{3}$.

Rationalising the denominator

Convention says a final answer should not have a surd in the denominator.

Why surds matter

Surds are how OCR asks for exact answers in Pythagoras, trigonometry with the special angles ($\sin 60^\circ = \tfrac{\sqrt{3}}{2}$) and the quadratic formula when the discriminant is not a perfect square. Keeping a value as $5\sqrt{2}$ rather than $7.07\ldots$ avoids rounding error that would otherwise compound through a multi-step problem, and OCR's mark schemes specifically reward exact surd answers where they are requested.