Question

Rationalize the denominator and simplify:
(i) $\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}$
(ii) $\frac{5+2\sqrt{3}}{7+4\sqrt{3}}$
(iii) $\frac{2\sqrt{6}-\sqrt{5}}{3\sqrt{5}-2\sqrt{6}}$
(iv) $\frac{4\sqrt{3}+5\sqrt{2}}{\sqrt{48}+\sqrt{18}}$

Answer 1

(i) $\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}$

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

$=\frac{(\sqrt{3}-\sqrt{2}{)}^2}{(\sqrt{3}{)}^2-(\sqrt{2}{)}^2}$

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

$=5-2\sqrt{6}$

(ii) $\frac{5+2\sqrt{3}}{7+4\sqrt{3}}$

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

$=\frac{35-20\sqrt{3}+14\sqrt{3}-24}{7^2-(4\sqrt{3}{)}^2}$

$=\frac{11-6\sqrt{3}}{49-48}=11-6\sqrt{3}$

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

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

$=\frac{6\sqrt{30}+24-15-2\sqrt{30}}{(3\sqrt{5}{)}^2-(2\sqrt{6}{)}^2}$

$=\frac{4\sqrt{30}+9}{45-24}$

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

(iv) $\frac{4\sqrt{3}+5\sqrt{2}}{\sqrt{48}+\sqrt{18}}$

$=\frac{4\sqrt{3}+5\sqrt{2}}{\sqrt{48}+\sqrt{18}}\times \frac{\sqrt{48}-\sqrt{18}}{\sqrt{48}-\sqrt{18}}$

$=(\frac{4\sqrt{3}+5\sqrt{2}\left)\right(4\sqrt{3}-3\sqrt{2})}{(\sqrt{48}{)}^2-(\sqrt{18}{)}^2}$

$=\frac{48-12\sqrt{6}+20\sqrt{6}-30}{48-18}$

$=\frac{18+8\sqrt{6}}{30}$

$=\frac{9+4\sqrt{6}}{15}$