If $$x=[\frac{1}{(\sqrt{5}+\sqrt{3})}], y=[\frac{1}{(\sqrt{7}+\sqrt{5})}]$$ and $$z=[\frac{1}{(\sqrt{7}+\sqrt{3})}]$$, then what is the value of (x+y+z) ?

Given : $$x=[\frac{1}{(\sqrt{5}+\sqrt{3})}]$$

Rationalizing the denominator, we get :

=> $$x=[\frac{1}{(\sqrt{5}+\sqrt{3})}]\times[\frac{(\sqrt5-\sqrt3)}{(\sqrt5-\sqrt3)}]$$

=> $$x=\frac{(\sqrt5-\sqrt3)}{(\sqrt5)^2-(\sqrt3)^2}$$

=> $$x=\frac{(\sqrt5-\sqrt3)}{2}$$

Similarly, $$y=\frac{(\sqrt7-\sqrt5)}{2}$$ and $$z=\frac{(\sqrt7-\sqrt3)}{4}$$

To find : $$x+y+z$$

= $$\frac{(\sqrt5-\sqrt3)}{2}$$ $$+\frac{(\sqrt7-\sqrt5)}{2}$$ $$+\frac{(\sqrt7-\sqrt3)}{4}$$

= $$\frac{\sqrt7}{2}+\frac{\sqrt7}{4}-\frac{\sqrt3}{2}-\frac{\sqrt3}{4}$$

= $$\frac{3\sqrt7}{4}-\frac{3\sqrt3}{4}=\frac{3(\sqrt7-\sqrt3)}{4}$$

=> Ans - (A)