Question

Simplify by rationalising the denominator.

(i) $\frac{7\sqrt{3}-5\sqrt{2}}{\sqrt{48}+\sqrt{18}}$

(ii) $\frac{2\sqrt{6}-\sqrt{5}}{3\sqrt{5}-2\sqrt{6}}$

Answer 1

(i)
$$\begin{gathered} \frac{7\sqrt{3}-5\sqrt{2}}{\sqrt{48}+\sqrt{18}} \\ =\frac{7\sqrt{3}-5\sqrt{2}}{\sqrt{16\times 3}+\sqrt{9\times 2}} \\ =\frac{7\sqrt{3}-5\sqrt{2}}{4\sqrt{3}+3\sqrt{2}} \end{gathered}$$
$$\begin{gathered} =\frac{7\sqrt{3}-5\sqrt{2}}{4\sqrt{3}+3\sqrt{2}}\times \frac{4\sqrt{3}-3\sqrt{2}}{4\sqrt{3}-3\sqrt{2}} \\ =\frac{7\sqrt{3}\times 4\sqrt{3}-7\sqrt{3}\times 3\sqrt{2}-5\sqrt{2}\times 4\sqrt{3}+5\sqrt{2}\times 3\sqrt{2}}{{\left(4\sqrt{3}\right)}^2-{\left(3\sqrt{2}\right)}^2} \\ =\frac{84-21\sqrt{6}-20\sqrt{6}+30}{48-18} \end{gathered}$$
$=\frac{114-41\sqrt{6}}{30}$
(ii)
$$\begin{gathered} \frac{2\sqrt{6}-\sqrt{5}}{3\sqrt{5}-2\sqrt{6}} \\ =\frac{2\sqrt{6}-\sqrt{5}}{3\sqrt{5}-2\sqrt{6}}\times \frac{3\sqrt{5}+2\sqrt{6}}{3\sqrt{5}+2\sqrt{6}} \\ =\frac{2\sqrt{6}\times 3\sqrt{5}+2\sqrt{6}\times 2\sqrt{6}-\sqrt{5}\times 3\sqrt{5}-\sqrt{5}\times 2\sqrt{6}}{{\left(3\sqrt{5}\right)}^2-{\left(2\sqrt{6}\right)}^2} \end{gathered}$$
$$\begin{gathered} =\frac{6\sqrt{30}+24-15-2\sqrt{30}}{45-24} \\ =\frac{9+4\sqrt{30}}{21} \end{gathered}$$