After rationalizing the denominator of $\frac{7}{3 \sqrt{3}-2 \sqrt{2}},$ we get the denominator as

Rationalise the denominator of the following number:
$\frac{1}{5+2 \sqrt{3}}$

Find which of the variables $x, y, z$ and $u$ represent rational numbers and which irrational numbers:
$(i)$ $x^{2}=5$
$(ii)$ $y^{2}=9$
$(iii)$ $z^{2}=0.04$
$(iv)$ $u^{2}=\frac{17}{4}$

If $\sqrt{2} = 1.4142$,then evaluate $\sqrt{5} \div \sqrt{10}$ correct to four decimal places.

Find the value of $a$ :
$\frac{3-\sqrt{5}}{3+2 \sqrt{5}}=a \sqrt{5}-\frac{19}{11}$

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