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Simplify the following: $\frac{4 \sqrt{3}+5 \sqrt{2}}{\sqrt{48}+\sqrt{18}}$.
Given: $\frac{4 \sqrt{3}+5 \sqrt{2}}{\sqrt{48}+\sqrt{18}}$.
To do: To simplify: $\frac{4 \sqrt{3}+5 \sqrt{2}}{\sqrt{48}+\sqrt{18}}$.
Solution:
As given $\frac{4\sqrt{3}+5\sqrt{2}}{\sqrt{48}+\sqrt{18}}$
$=\frac{4\sqrt{3}+5\sqrt{2}}{4\sqrt{3}+3\sqrt{2}}$ [$\because \sqrt{48}=4\sqrt{3}$ and $\sqrt{18}=3\sqrt{2}$]
On multiplying both numerators and denominators both by $4\sqrt{3}-3\sqrt{2}$
$=\frac{4\sqrt{3}+5\sqrt{2}}{4\sqrt{3}+3\sqrt{2}}\times\frac{4\sqrt{3}-3\sqrt{2}}{4\sqrt{3}-3\sqrt{2}}$
$=\frac{( 4\sqrt{3}+5\sqrt{2})(4\sqrt{3}-3\sqrt{2})}{( 4\sqrt{3})^2-( 3\sqrt{2})^2}$
$=\frac{48+20\sqrt{6}-12\sqrt{6}-30}{48-18}$
$=\frac{18+8\sqrt{6}}{30}$
$=\frac{2( 9+4\sqrt{6}}{30}$
$=\frac{9+4\sqrt{6}}{15}$
Thus, $\frac{4 \sqrt{3}+5 \sqrt{2}}{\sqrt{48}+\sqrt{18}}=\frac{9+4\sqrt{6}}{15}$.
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