Question

Prove that root 6 is irrational please its very urgent answer it fast

Answer 1

let us assume that √6 is a rational no.
√6 = p/q (where p , q are co primes , q is not equal to zero)
(√6)^2 =(p/q)^2. (squaring both sides)
6 = p^2/q^2
p^2 = 6q^2 ----------(1)
p^2 is divisible by 6
and p is also divisible by 6
p = 6m
substituting values in 1
(6m)^2 = (6q^2)
36m^2 = 6q^2
6m^2 = q^2 (taking 6 to LHS)
q^2 = 6m^2
q^2 is divisible by 6

therefore p and q have atleast one common factor 6 . this contradicts to the fact that p and q have no common factor . this is becoz our assumption is wrong .
therefore √6 is an irrational number ....hence proved!