Question

prove that 3 root 3 is an irrational numbers​

Answer 1

Question:

Prove that $3\sqrt{3}$ is an irrational Number.

Answer:

Let's assume that $3\sqrt{3}$ is a rational number. Then, there exists positive integers a and b such that,

ㅤㅤㅤㅤ$:\implies$ ${3\sqrt{3}=\frac{a}{b}}$

ㅤㅤㅤㅤ$:\implies$ $\frac{3\sqrt{3}}{3}=\large\frac{a}{3b}$

ㅤㅤㅤㅤ$:\implies$ $\sqrt{3}=\large\frac{a}{3b}$

We know, $\large\frac{a}{3b}$ is a rational number therefore, $\sqrt{3}$ is also a rational number.

But, this contradicts the fact that $\sqrt{3}$ is an irrational Number.

Therefore, $\mathsf{3\sqrt{3}}$ is irrational number

Hence Proved!!