# Simplify by rationalising the denominator.

Question:

Simplify by rationalising the denominator.

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

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

Solution:

(i)

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

$=\frac{7 \sqrt{3}-5 \sqrt{2}}{\sqrt{16 \times 3}+\sqrt{9 \times 2}}$

$=\frac{7 \sqrt{3}-5 \sqrt{2}}{4 \sqrt{3}+3 \sqrt{2}}$

$=\frac{7 \sqrt{3}-5 \sqrt{2}}{4 \sqrt{3}+3 \sqrt{2}} \times \frac{4 \sqrt{3}-3 \sqrt{2}}{4 \sqrt{3}-3 \sqrt{2}}$

$=\frac{7 \sqrt{3} \times 4 \sqrt{3}-7 \sqrt{3} \times 3 \sqrt{2}-5 \sqrt{2} \times 4 \sqrt{3}+5 \sqrt{2} \times 3 \sqrt{2}}{(4 \sqrt{3})^{2}-(3 \sqrt{2})^{2}}$

$=\frac{84-21 \sqrt{6}-20 \sqrt{6}+30}{48-18}$

$=\frac{114-41 \sqrt{6}}{30}$

(ii)

$\frac{2 \sqrt{6}-\sqrt{5}}{3 \sqrt{5}-2 \sqrt{6}}$

$=\frac{2 \sqrt{6}-\sqrt{5}}{3 \sqrt{5}-2 \sqrt{6}} \times \frac{3 \sqrt{5}+2 \sqrt{6}}{3 \sqrt{5}+2 \sqrt{6}}$

$=\frac{2 \sqrt{6} \times 3 \sqrt{5}+2 \sqrt{6} \times 2 \sqrt{6}-\sqrt{5} \times 3 \sqrt{5}-\sqrt{5} \times 2 \sqrt{6}}{(3 \sqrt{5})^{2}-(2 \sqrt{6})^{2}}$

$=\frac{6 \sqrt{30}+24-15-2 \sqrt{30}}{45-24}$

$=\frac{9+4 \sqrt{30}}{21}$