If $$x=7+2\sqrt{10}$$, then what is the value of $$(\sqrt{x}-\frac{1}{\sqrt{x}})$$

Expression : $$x=7+2\sqrt{10}$$

=> $$x=(\sqrt5)^2+(\sqrt2)^2+2(\sqrt5)(\sqrt2)$$

Using, $$a^2+b^2+2ab=(a+b)^2$$

=> $$x=(\sqrt5+\sqrt2)^2$$

=> $$\sqrt{x}=\sqrt5+\sqrt2$$ -------------(i)

Also, $$\frac{1}{\sqrt{x}}=\frac{1}{\sqrt5+\sqrt2}$$

Rationalizing the denominator, we get :

=> $$\frac{1}{\sqrt{x}}=\frac{1}{\sqrt5+\sqrt2}\times(\frac{\sqrt5-\sqrt2}{\sqrt5-\sqrt2})$$

=> $$\frac{1}{\sqrt{x}}=\frac{\sqrt5-\sqrt2}{5-2}$$

=> $$\frac{1}{\sqrt{x}}=\frac{(\sqrt5-\sqrt2)}{3}$$ ---------(ii)

Subtracting equation (ii) from (i),

$$\therefore$$ $$(\sqrt{x}-\frac{1}{\sqrt{x}})=(\sqrt5+\sqrt2)-(\frac{\sqrt5-\sqrt2}{3})$$

= $$\frac{2\sqrt5}{3}+\frac{4\sqrt2}{3}$$

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

=> Ans - (D)