Simplify the expression by rationalising the denominator: $\frac{5}{\sqrt{7}-\sqrt{2}}$.

Step-by-step solution

1. Identify the conjugate (rationalising factor) of the denominator $\sqrt{7}-\sqrt{2}$, which is $\sqrt{7}+\sqrt{2}$.
2. Multiply both the numerator and the denominator by $(\sqrt{7}+\sqrt{2})$: $\frac{5}{\sqrt{7}-\sqrt{2}}\times \frac{\sqrt{7}+\sqrt{2}}{\sqrt{7}+\sqrt{2}}$.
3. Use the algebraic identity $(a-b)(a+b)=a^2-b^2$ in the denominator: $\frac{5(\sqrt{7}+\sqrt{2})}{(\sqrt{7}{)}^2-(\sqrt{2}{)}^2}$.
4. Simplify the denominator: $(\sqrt{7}{)}^2-(\sqrt{2}{)}^2=7-2=5$.
5. Cancel out the common factor $5$ from both the numerator and the denominator: $\frac{5(\sqrt{7}+\sqrt{2})}{5}=\sqrt{7}+\sqrt{2}$.

Concept tested: Rationalising the denominator using identities

Published 20 August 2026 · How this solution was prepared and checked

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