Expressing $\frac{1}{\sqrt{2} + \sqrt{3} + \sqrt{5}}$ with rational Denominator

Question

could you please help me express this with a rational denominator

$\frac{1}{\sqrt{2} + \sqrt{3} + \sqrt{5}}$

Thank you

Answer 1

Hint:

First, multiply the expression with $$\frac{(\sqrt 2 + \sqrt 3) - \sqrt 5}{(\sqrt 2 + \sqrt 3) - \sqrt 5}$$

What do you get as a result? Is the way forward clear to you? If not, post the result and we can see what to do in the next step.

Answer 2

We can use the conjugate trick for rationalizing denominators more than once:

$$\frac1{\sqrt2+\sqrt3+\sqrt5}\frac{\sqrt2+\sqrt3-\sqrt5}{\sqrt2+\sqrt3-\sqrt5}\frac{\sqrt6}{\sqrt6}=\frac{\sqrt2+\sqrt3-\sqrt5}{2\sqrt6}\frac{\sqrt6}{\sqrt6}=\frac{2\sqrt3+3\sqrt2-\sqrt{30}}{12}$$

Answer 3

$$\frac{1}{\sqrt{2} + \sqrt{3} + \sqrt{5}}=\frac{1}{\sqrt{2} + \sqrt{3} + \sqrt{5}}\frac{\sqrt{2} + \sqrt{3} - \sqrt{5}}{\sqrt{2} + \sqrt{3}-\sqrt{5}}=$$ $$=\frac{\sqrt{2} + \sqrt{3} - \sqrt{5}}{2\sqrt6}=\frac{\sqrt{2} + \sqrt{3} - \sqrt{5}}{2\sqrt6}\frac{\sqrt6}{\sqrt6}=\frac{\sqrt{12}+\sqrt{18}-\sqrt{30}}{12}$$