Simplify the following:
$4 \sqrt{12} \times 7 \sqrt{6}$

If $\sqrt{2} = 1.4142$,then $\sqrt{\frac{\sqrt{2}-1}{\sqrt{2}+1}}$ is equal to

State whether the following statements are true or false. Justify your answer.
$(i)$ The number of rational numbers between $15$ and $18$ is finite.
$(ii)$ There are numbers which cannot be written in the form $\frac{p}{q}$,where $q \neq 0$ and $p, q$ are both integers.

Express the following in the form $\frac{p}{q},$ where $p$ and $q$ are integers and $q \neq 0$ :
$0 . \overline{001}$

Simplify the following:
$(\sqrt{3}-\sqrt{2})^{2}$

Classify the following numbers as rational or irrational with justification:
$(i)$ $\sqrt{\frac{9}{27}}$
$(ii)$ $\frac{\sqrt{28}}{\sqrt{343}}$