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

Surds are a Higher-tier CCEA Number topic that keeps answers exact rather than rounded. A surd is a root that cannot be written exactly as a fraction. You must simplify surds, add, subtract, multiply and divide them, expand brackets containing them, and rationalise denominators, including those of the form $a + \sqrt{b}$. Surds appear on the non-calculator work and feed directly into Pythagoras, exact-value trigonometry and the quadratic formula, 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 $\sqrt{2}, \sqrt{3}$ and $\sqrt{5}$ are surds, but $\sqrt{9} = 3$ is not. Leaving an answer as a surd keeps it exact, which is why CCEA asks for answers "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, such as $\sqrt{72} = \sqrt{4 \times 18} = 2\sqrt{18}$, 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 CCEA asks for exact answers in Pythagoras, in trigonometry with the special angles such as $\sin 60^\circ = \tfrac{\sqrt{3}}{2}$, and in 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 build up through a multi-step problem, and the mark schemes specifically reward exact surd answers where they are requested.