Question

show that 3 root 3 is an irrational number

Answer 1

Let us assume, the contrary that 3√3 is a rational.
I. e., we can find coprimes a and b (b ≠0) such that
$3 \sqrt{3 } = \frac{a \:}{b}$
we get
$\sqrt{3 } = \frac{a}{3b}$
since 3,a and b are integers,
$\frac{a}{3b}$

is rational, and so √3 is a rational.
But this contradicts the fact that √3 is an irrational.
so we conclude that 3√3 is an irrational