Question

Give an example of a number x such that x² is an irrational number and x³ is a rational number.

Answer 1

The cube roots of natural numbers which are not perfect cubes are all irrational numbers.
Let $x=\sqrt[3]{2}=2^{\frac{1}{3}}$.
Now,
$x^2={\left(2^{\frac{1}{3}}\right)}^2=2^{\frac{2}{3}}={\left(2^2\right)}^{\frac{1}{3}}=4^{\frac{1}{3}}$, which is an irrational number
Also,
$x^3={\left(2^{\frac{1}{3}}\right)}^3=2^{3\times \frac{1}{3}}=2$, which is a rational number