Visualise the representation of $4.\overline{67}$ on the number line up to $4$ decimal places (round up the number to correct four decimal places).

If the rationalizing factor of the denominator $\frac{1}{\sqrt{2}+\sqrt{3}}$ is in the form of $\sqrt{k}-\sqrt{\left(k+1\right)}$, then find the value of $k$.

If $\frac{1}{\sqrt{7}}=\frac{\sqrt{a}}{b}$, then find the value of $\left(a+b\right)$.

If $\frac{\sqrt{5}}{2\sqrt{3}}=\frac{\sqrt{a}}{b}$, then find the value of $a+b$.

If the rationalised form of $\frac{1}{2+\sqrt{3}}$ is of the form $a\pm \sqrt{b}$, then find the value of $\left(a+b\right)$.

If $\frac{1}{\sqrt{5}-2}=\sqrt{a}+b$, then find the value of $a+b$.

If the rationalized form of $\frac{1}{5+3\sqrt{2}}$ is $\frac{k-m\sqrt{n}}{r},$ then find the value of $k+m+n+r$.

If the rationalized form of  $\frac{1}{\sqrt{7}-\sqrt{6}}$ is $\sqrt{a}+\sqrt{b}$, then find the value of $a+b$.

When we Rationalise the denominator of .$\frac{4}{\sqrt{11}-\sqrt{7}}$ we get $\sqrt{a}+\sqrt{b}$. Find the value of $a$