Question

Find two irrational numbers lying between $\sqrt{2}and\sqrt{3}$.

Answer 1

We know that, if a and b are two distinct positive irrational numbers, then $\sqrt{ab}$ is an irrational number lying between a and b, if ab is not a perfect square
$\therefore$ Irrational number between $\sqrt{2}and\sqrt{3}is\sqrt{\sqrt{2}\times \sqrt{3}}=\sqrt{\sqrt{6}}=6^{1/4}$
Irrational number between $\sqrt{2}and{6}^{1/4}is\sqrt{\sqrt{2}\times 6^{1/4}}=2^{1/4}\times 6^{1/8}=2^{2/8}\times 6^{1/8}$
$=(4\times 6{)}^{1/8}={24}^{1/8}$
Hence required irrational numbers are $6^{1/4}and{24}^{1/8}$