The product of any two irrational numbers is

If $a = 2 + \sqrt{3}$,then find the value of $a - \frac{1}{a}$.

Rationalise the denominator in each of the following:
$\frac{3+2 \sqrt{2}}{3-2 \sqrt{2}}$

Simplify:
${{(625)^{-\frac{1}{2}}}^{-\frac{1}{4}}}^{2}$

Simplify:
$\frac{9^{\frac{1}{3}} \times 27^{-\frac{1}{2}}}{3^{\frac{1}{6}} \times 3^{-\frac{2}{3}}}$

State whether the following statement is true:
There is a number $x$ such that $x^{2}$ is irrational but $x^{4}$ is rational. Justify your answer by an example.