Question

Give an example for each of the following:
(i) The product of two irrational numbers is a rational number.
(ii) The product of two irrational numbers is an irrational number.

Answer 1

1- The product of two irrational numbers is an irrational number.
The numbers $\sqrt{2}$ and $\sqrt{3}$ are both irrational and so $\sqrt{2}.\sqrt{3}$ = $\sqrt{2.3}$ = $\sqrt{6}$
So the product of $\sqrt{2}$ and $\sqrt{3}$ both irrational is $\sqrt{6}$ (which is also irrational)
2- The product of two irrational numbers is a rational number.
The numbers $\sqrt{2}$ and $\sqrt{8}$ are irrational , but $\sqrt{2}$.$\sqrt{8}$ = $\sqrt{16}$ = $4$ is rational number.
So the product of $\sqrt{2}$ and $\sqrt{8}$ which are irrational numbers is $4$ (rational number)