Question

prove that √x is not a rational number if x is not perfect square

Answer 1

Let us suppose that √5 is a rational number.

So √5 = p/q

=> 5 = p2 /q2

=>5q2 = p2 ..............1

So p2 is divisible by 5

=> p is divisible by 5

Let p =5x  (x is a positive integer)

Now p2  = 25c2

from equation 1

5q2 = 25c2

=> q2 = 5c2

So q is divisible by 5

Thus p and q has a common factor 5. It is contradiction of our assumption.

So, √5 is not a rational number.

use x in the place of 5