Surds: simplifying surds, the four operations with surds, expanding brackets containing surds, and rationalising the denominator (Higher tier).

What this dot point is asking

A surd is a root that cannot be simplified to a rational number, such as $\sqrt{2}$ or $\sqrt{3}$. Surds are a Higher-tier topic, and Edexcel expects you to simplify them, calculate with them while keeping answers exact, expand brackets containing surds, and rationalise a denominator so no root is left on the bottom. Working in surds keeps answers exact, which is why questions often say "give your answer in surd form".

Simplifying surds

The key rule is that a root of a product splits into a product of roots, which lets you pull out perfect squares.

To simplify $\sqrt{72}$, look for the largest square factor: $72 = 36 \times 2$, so $\sqrt{72} = \sqrt{36}\,\sqrt{2} = 6\sqrt{2}$. Choosing the largest square factor saves steps; if you only spot $\sqrt{72} = \sqrt{4}\,\sqrt{18} = 2\sqrt{18}$, you then have to simplify $\sqrt{18}$ again.

Adding, subtracting and multiplying surds

Surds behave like algebraic terms. You can add or subtract only when the surd parts match: $3\sqrt{5} + 4\sqrt{5} = 7\sqrt{5}$, but $3\sqrt{5} + 4\sqrt{2}$ cannot be combined. Often you must simplify first to reveal matching surds, as in the worked questions above.

For multiplication, multiply the rational parts and the surd parts separately: $2\sqrt{3} \times 5\sqrt{2} = 10\sqrt{6}$. A vital special case is $\sqrt{a} \times \sqrt{a} = a$, so $\sqrt{7} \times \sqrt{7} = 7$.

Expanding brackets with surds

Expand exactly as you would with algebra, then simplify any surd products. For $(2 + \sqrt{3})(5 - \sqrt{3})$, multiply each pair of terms:

Note that $\sqrt{3} \times \sqrt{3} = 3$, and the middle terms combine because they share $\sqrt 3$.

Rationalising the denominator

A surd in the denominator is considered "untidy", so you rationalise it: remove the root from the bottom while keeping the value unchanged.

For a single surd on the bottom, just multiply by that surd: $\dfrac{10}{\sqrt{2}} = \dfrac{10}{\sqrt{2}} \times \dfrac{\sqrt{2}}{\sqrt{2}} = \dfrac{10\sqrt{2}}{2} = 5\sqrt{2}$.

Why surds matter elsewhere

Surds are not an isolated topic; they appear whenever an exact answer is required. When you solve a quadratic with the formula, the discriminant is usually not a perfect square, so the roots come out as surds, for example $x = \dfrac{3 \pm \sqrt{5}}{2}$. When you use Pythagoras or trigonometry, exact side lengths are often surds, such as the diagonal of a unit square being $\sqrt{2}$, or the exact value $\sin 60^\circ = \dfrac{\sqrt{3}}{2}$. Leaving these in surd form, rather than rounding, keeps later working accurate and is what "give an exact answer" demands. Because of this, a question that looks like geometry or algebra can hide a surd-simplification mark, so the fluency you build here is reused across the Higher paper.

Try this

Q1. Simplify fully $\sqrt{8} \times \sqrt{6}$. [2 marks]

• Cue. $\sqrt{8}\,\sqrt{6} = \sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$.

Q2. Rationalise the denominator of $\dfrac{5}{\sqrt{5}}$. [2 marks]

• Cue. Multiply top and bottom by $\sqrt{5}$: $\dfrac{5\sqrt{5}}{5} = \sqrt{5}$.