Question

Find the value of $x$ and $y$, if $\frac{\sqrt{2}}{3\sqrt{6}-\sqrt{5}}=x\sqrt{3}-y\sqrt{10}$.

Answer 1

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

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

$=\frac{3\sqrt{2}\sqrt{2\times 3}+\sqrt{10}}{54-5}$

$=\frac{3\times 2\sqrt{3}+\sqrt{10}}{49}$

$=\frac{6\sqrt{3}+\sqrt{10}}{49}=x\sqrt{3}-y\sqrt{10}$

$x=\frac{6}{49}$ $y=\frac{-1}{49}$