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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Updated on: 10-Oct-2022

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