Rationalise the denominator of $\frac{1}{\sqrt{2}}$.

Show that $3.142678$ is a rational number. In other words,express $3.142678$ in the form $\frac{p}{q}$,where $p$ and $q$ are integers and $q \ne 0$.

Simplify:
$(i)$ $2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}}$
$(ii)$ $\left(3^{\frac{1}{5}}\right)^{4}$
$(iii)$ $\frac{7^{\frac{1}{5}}}{7^{\frac{1}{3}}}$
$(iv)$ $13^{\frac{1}{5}} \cdot 17^{\frac{1}{5}}$

Find six rational numbers between $3$ and $4$.

Find the values of:
$(i)$ $64^{\frac{1}{2}}$
$(ii)$ $32^{\frac{1}{5}}$
$(iii)$ $125^{\frac{1}{3}}$

Express the following in the form $\frac{p}{q}$,where $p$ and $q$ are integers and $q \ne 0$.
$(i)$ $0.\overline{6}$
$(ii)$ $0.4\overline{7}$
$(iii)$ $0.\overline{001}$