If $x$ and $y$ are real numbers satisfying $\dfrac{\sqrt{x}+\sqrt{y}}{x-y}-\dfrac{\sqrt{y}-\sqrt{x}}{x+y}=4$, then find the value of $m$ if $\dfrac{{{x}^{m}}+{{y}^{m}}}{{{x}^{2}}-{{y}^{2}}}=2$
Solution
$\dfrac{\sqrt{x}+\sqrt{y}}{x-y}-\dfrac{\sqrt{y}-\sqrt{x}}{x+y}=4$
$\Rightarrow \dfrac{(x+y)(\sqrt{x}+\sqrt{y})-(\sqrt{y}-\sqrt{x}(x-y)}{{{x}^{2}}-{{y}^{2}}}=4$
$\Rightarrow \dfrac{2x \sqrt{x}+2y\sqrt{y}}{{{x}^{2}}-{{y}^{2}}}=4$
$\Rightarrow \dfrac{{{x}^{\frac{3}{2}}}+{{y}^{\frac{3}{2}}}}{{{x}^{2}}-{{y}^{2}}}=2$
Hence, $m=\dfrac{3}{2}$