prove that there is no rational no. whose squre is 8.

Answer :
Let assume that number is rational , whose square is 8 ,
Hence there are two integers p and q , being co-prime such that 

${\left( \frac{p}{q} \right)}^2$  = 8 

p² = 8q^​2

p² = 2 ( 2q)² 

Hence p must have prime factor of 2  ,
So we substitute p = 2k  , where k is some integer ,So
( 2 k )² = 2 ( 2q )²
4k² = 2 ( 2q)²

2 k² = ( 2q )²
So,
Hence q must have prime factor 2 , So H.C.F. ( p , q ) $\ge$ 2 , that contradict our assumption ,So

There is no rational number whose square is 8   .                 ( Hence proved )