State whether the following statements are true or false. Justify your answer.
$(i)$ The square of an irrational number is always rational.
$(ii)$ $\frac{\sqrt{12}}{\sqrt{3}}$ is not a rational number as $\sqrt{12}$ and $\sqrt{3}$ are not integers.

(N/A) $(i)$ The statement is false. Consider the irrational number $\sqrt[4]{2}$. Its square is $(\sqrt[4]{2})^{2} = \sqrt{2}$,which is an irrational number.
$(ii)$ The statement is false. We can simplify the expression as $\frac{\sqrt{12}}{\sqrt{3}} = \sqrt{\frac{12}{3}} = \sqrt{4} = 2$. Since $2$ can be written as $\frac{2}{1}$,it is a rational number.