Operations on Real Numbers

Simplify, $\left(3+\sqrt{23}\right)-\sqrt{23}$

Solve: $\frac{7}{\sqrt{5}-\sqrt{3}}$.

Simplify : $\frac{2\sqrt{27}+3\sqrt{12}}{4\sqrt{3}}$

Simplify the following expressions:

$\left(5+\sqrt{7}\right)\left(2+\sqrt{5}\right)$

Given that $\frac{2-\sqrt{5}}{2+\sqrt{5}}=a\sqrt{5}+b$, find $a$ and $b$.

Write two rationalising factors of $(\sqrt{15}+\sqrt{3})$.

Find: $\frac{\sqrt{10}+\sqrt{15}}{\sqrt{2}}$

Simplify by rationalisation the denominator in $\frac{3\sqrt{5} + \sqrt{7}}{3\sqrt{5}-\sqrt{7}}$.

Find :$\frac{1}{\sqrt{5}+\sqrt{7}}+\frac{1}{\sqrt{9}{\displaystyle +}{\displaystyle \sqrt{11}}}+...+\frac{1}{\sqrt{9637}{\displaystyle +}{\displaystyle \sqrt{9639}}}$.

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

Find $\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}$.

Prove that: $\frac{1}{2+\sqrt{3}}+\frac{2}{\sqrt{5}-\sqrt{3}}+\frac{1}{2-\sqrt{5}}=0$

Simplify:

$\left(2\sqrt{5}-3\sqrt{2}\right)\left(3\sqrt{5}+7\sqrt{2}\right)$

Rationalise $\frac{3+\sqrt{2}}{5+\sqrt{2}}$.

If $a=1-\sqrt{2}\text{ find }{\left(a-\frac{1}{a}\right)}^3$

Simplify:$\frac{7+7\sqrt{5}}{3+\sqrt{5}}-\frac{7-7\sqrt{5}}{3-\sqrt{5}}$

Simplify :$2\sqrt{3}-\frac{\sqrt{2}}{2\sqrt{3}}+\sqrt{2}$

If $x=2+\sqrt{5}$, then find the value of ${\left(x-\frac{1}{x}\right)}^3$

Find the value:

$\frac{4}{3\sqrt{3}-2\sqrt{2}}+\frac{3}{3\sqrt{3}+2\sqrt{2}}$

If $a=\frac{4}{3-\sqrt{5}}$, find $a+\frac{4}{a}$