Question

to prove that 6√2 is irrational.

Answer 1

Assuming that  is rational. Then,  where   and  are integers and  is in lowest terms. This means that  and  cannot be both even  is irrational.
Squaring both sides, we have 

Multiplying both sides by , we have . It follows that  is even and   is even.
If  is even, then it can be expressed as  where  is an integer. Substituting to the equation above, we have  which simplifies to.Dividing both sides by  gives . This implies that  is even which means that  is even. 
similarly  is irrational.
therefore  6 is totally an irrational number .

Answer 2

let 6 √ 2 be an rational no.
6 √ 2 = x/y (x,y are integers , y not equal to zero)
√ 2 = x/6y
LHS^    RHS^

LHS = √ 2 = irrational no.
RHS = x/6y = rational no.

but this is a contradiction.....any irrational no. cannot be equal to rational no.
therefore, 6 √ 2 is an irrational no.