The decimal expansion of the number $\sqrt{2}$ is

Insert a rational number and an irrational number between the following:

$\sqrt{2}$ and $\sqrt{3}$

Simplify the following expressions

$(\sqrt{15}+\sqrt{7})(\sqrt{15}-\sqrt{7})$

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

Simplify: $\frac{7 \sqrt{3}}{\sqrt{10}+\sqrt{3}}-\frac{2 \sqrt{5}}{\sqrt{6}+\sqrt{5}}-\frac{3 \sqrt{2}}{\sqrt{15}+3 \sqrt{2}}$

Express $2 . \overline{137}$ in the form $\frac{p}{q} ;$ where $p$ and $q$ are integers and $q \neq 0$