Question

Give an example of an irrational number such that its $8^{th}$ power is a rational number.

Answer 1

Let the irrational number be $\sqrt[8]{82}$ that is $2^{\frac{1}{8}}$.

Then the $8^{th}$ power will be as follows:

${\left(2^{\frac{1}{8}}\right)}^8=2^{\frac{1}{8}\times 8}=2^1=2(\because {\left(a^x\right)}^y=a^{xy})$

Hence, $2$ is a rational number.