Question

Let $x$ be rational and $y$ be irrational. Is $xy$ necessarily irrational$?$ Justify your answer by an example.

Answer 1

Determine whether the product of $x$ and $y$ is an irrational number:

As we know, the product of a rational and irrational number is always an irrational number.

So, if $x$ and $y$ be rational and irrational numbers respectively.

Then $xy$ necessarily an irrational number.

For example: assume that $x=1$and $y=\sqrt{7}$.

Then, $xy=\sqrt{7}$ which is an irrational number.

Hence, $xy$ is necessarily an irrational number.