Find the values of $a$ and $b$ in each of the following:

$\frac{7+\sqrt{5}}{7-\sqrt{5}}-\frac{7-\sqrt{5}}{7+\sqrt{5}}=a+\frac{7}{11} \sqrt{5} b$

Classify the following numbers as rational or irrational with justification:

$(i)$ $\sqrt{\frac{9}{27}}$

$(ii)$ $\frac{\sqrt{28}}{\sqrt{343}}$

Find the value

$64^{\frac{5}{6}}$

Rationalise the denominator in each of the following

$\frac{3+2 \sqrt{2}}{3-2 \sqrt{2}}$

Prove that, $\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}=1$

Simplify the following:

$\sqrt{45}-3 \sqrt{20}+4 \sqrt{5}$