Rationalising the Denominator

If $a$ and $b$ are two rational numbers then find the value of $a$and $b$ in following equation

$\frac{6+\sqrt{3}}{6-\sqrt{3}}=a+b\sqrt{3}$

Simplify

$\frac{2+\sqrt{3}}{2-\sqrt{3}}+\frac{2-\sqrt{3}}{2+\sqrt{3}}+\frac{\sqrt{3}+1}{\sqrt{3}-1}$

Simplify

$\frac{5+\sqrt{3}}{2-\sqrt{3}}+\frac{2-\sqrt{3}}{2+\sqrt{3}}$

If $a$ and $b$ are two rational numbers then find the value of $a$and $b$ in following equation

$\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}=a+b\sqrt{15}$

Simplify: ${\left(\sqrt{7}-\sqrt{3}\right)}^2$

If the rationalised denominator of $\frac{1}{\sqrt{5}}$ is $\frac{\sqrt{5}}{\mathrm{k}}$ then write the value of $\mathrm{k}$.

If $x=3-2\sqrt{2}$ then find the value of $x+\frac{1}{x}$

Simplify: $\left(5+\sqrt{5}\right)\left(5-\sqrt{5}\right)$

Simplify: $\left(\sqrt{2}+\sqrt{3}\right)\left(\sqrt{3}+\sqrt{8}\right)$

If the Rationalised denominator of $\frac{\sqrt{21}}{\sqrt{3}}$ is $\frac{\sqrt{63}}{\mathrm{k}}$, then write the value of $\mathrm{k}$.

If the rationalised denominator of $\frac{2}{\sqrt{6}}$is $\frac{\mathrm{k}\sqrt{6}}{6}$then find the value of $\mathrm{k}$.