Question

Rationalise the denominator and simplify:

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

Answer 1

Given: $\frac{4\sqrt{3}+5\sqrt{2}}{\sqrt{48}+\sqrt{18}}$

Multiplying both numerator and denominator by $\sqrt{48}-\sqrt{18}$ to rationalise the denominator.

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

$=\frac{48-12\sqrt{6}+20\sqrt{6}-30}{(\sqrt{48}{)}^2-(\sqrt{18}{)}^2}$

$=\frac{18+8\sqrt{6}}{48-18}$

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

$=\frac{2(9+4\sqrt{6})}{30}$

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