Question

check whether the following statement is True or False:

The product of these two irrational numbers $\sqrt{3}+3\sqrt{2}$ and $\sqrt{2}-\sqrt{3}$ is also an irrational number.

Answer 1

To get the product as irrational, the numbers need to have their rational and irrational parts different such that their product too has an irrational part.
The given pair of numbers are $3+3\sqrt{2}$ and $\sqrt{2}-\sqrt{3}$.
Multiplying these 2 numbers we get,
$=(3+3\sqrt{2})\times (\sqrt{2}-\sqrt{3})$
$=3\times (\sqrt{2}-\sqrt{3})+3\sqrt{2}\times (\sqrt{2}-\sqrt{3})$
$=3\sqrt{2}-3\sqrt{3}+6-3\sqrt{6}$
It is an irrational number.