GCSE · Surds

Expanding brackets with surdsExam-style questions

Why surds multiply out exactly like algebra — with √a·√a collapsing to a — how squares like (4 − √2)² keep their middle term, and why conjugate pairs multiply to a whole number.

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10 questions · GCSE (Higher) level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 3 Level 4 Level 5

1
Expand and simplify $(2 + \sqrt{5})(3 - \sqrt{5})$ .
Level 4 · 3 marksTry it
2
Write $(4 - \sqrt{2})^2$ in the form $a + b\sqrt{2}$ , where $a$ and $b$ are integers.
Level 4 · 2 marksTry it
3
Show that $(5 + \sqrt{3})(5 - \sqrt{3}) = 22$ .
Level 4 · 2 marksTry it
4
Write $(\sqrt{7} + 1)^2$ in the form $a + b\sqrt{7}$ , where $a$ and $b$ are integers.
Level 4 · 2 marksTry it
5
Expand and simplify $(2\sqrt{3} + 1)(\sqrt{3} - 4)$ .
Level 4 · 3 marksTry it

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