Question

prove that root 2 root 5 is irrational

Answer 1

Answer.

let 2.root 5 be rational
then it must in the form of p/q [q is not equal to 0][p and q are co-prime]
2root 5=p/q
=>2 root 5 * q = p
squaring on both sides
=> 20*q*q = p*p  ------> 1
p*p is divisible by 20
p is divisible by 20
p = 20c  [c is a positive integer] [squaring on both sides ]
p*p = 400c*c  --------- > 2
sub p*p in 1
20*q*q = 00*c*c
q*q = 20*c*c
=> q is divisble by 20
thus q and p have a common factor 20
there is a contradiction
as our assumsion p &q are co prime but it has a common factor
so 2√5 is an irrational