Find value of $\frac{x+{\sqrt5}}{x-{\sqrt5}}+\frac{x+{\sqrt3}}{x-{\sqrt3}}$

Question

$$\mbox{If } x=\frac{2{\sqrt {15}}}{{\sqrt5} + {\sqrt3}}$$ Find value of $$\frac{x+{\sqrt5}}{x-{\sqrt5}}+\frac{x+{\sqrt3}}{x-{\sqrt3}}$$

I tried with conjugate with value of $x$ which results in ${\sqrt{15}}({\sqrt 5}-{\sqrt 3})$ but could not figure out how to proceed after substituting that value in the required equation

Answer 1

HINT: The value of $x$ can be simplified as $$x=\sqrt{15}(\sqrt5-\sqrt3)=5\sqrt3-3\sqrt5$$ Substitute the value of $x$ in the expression $$\frac{5\sqrt3-3\sqrt5+\sqrt5}{5\sqrt3-3\sqrt5-\sqrt5}+\frac{5\sqrt3-3\sqrt5+\sqrt3}{5\sqrt3-3\sqrt5-\sqrt3}$$ $$=\frac{5\sqrt3-2\sqrt5}{5\sqrt3-4\sqrt5}+\frac{6\sqrt3-3\sqrt5}{4\sqrt3-3\sqrt5}$$ Multiply denominators by conjugates to rationalise fractions and simplify to get the value

Answer 2

Thanks Harish

to complete the remaining

multiplying with conjugate gives

$$\frac{35+10\sqrt{15}}{-5} + \frac{27+6\sqrt{15}}{3}$$

after cross multiplication $$ \frac {-30}{-15} = 2$$