Question

Prove 3 root 7 is not a rational number.

Answer 1

let 3√7 be rational
then 3√7=a/b where a and b are integers
√7 =a/3b
since a and b are Integers therefore a/3b is rational but this contradicts the fact that √7 is irrational

hence 3√7 is irrational

Answer 2

Answer:

We prove that 3√7 is not rational no. .i.e., it is an irrational no.

let us assume 3√7 is  rational

⇒   $3\sqrt{7}=\frac{a}{b}$  where a , b are some integer

Now,  

$3\sqrt{7}=\frac{a}{b}$

$\sqrt{7}=\frac{a}{b}\times\frac{1}{3}$

$\sqrt{7}=\frac{a}{3b}$

In RHS 3 , a , b are integers which given RHS is rational

⇒ LHS is also rational .i.e., √7 is rational

but this contradict the fact that √7 is irrational

So, this contradiction arise due to our wrong assumption

⇒ 3√7 is not rational no. .i.e., 3√7 is an irrational no.

Hence Proved