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

Show that $0.3333... = 0.\overline{3}$ can be expressed in the form $\frac{p}{q}$,where $p$ and $q$ are integers and $q \ne 0$.

Simplify the following expressions :
$(i)$ $(5+\sqrt{7})(2+\sqrt{5})$
$(ii)$ $(5+\sqrt{5})(5-\sqrt{5})$
$(iii)$ $(\sqrt{3}+\sqrt{7})^{2}$
$(iv)$ $(\sqrt{11}-\sqrt{7})(\sqrt{11}+\sqrt{7})$

Are the square roots of all positive integers irrational? If not,give an example of the square root of a number that is a rational number.

Multiply $6 \sqrt{5}$ by $2 \sqrt{5}$.

Express $0.99999 \ldots$ in the form $\frac{p}{q}$. Are you surprised by your answer? With your teacher and classmates,discuss why the answer makes sense.