Question

Prove that
$\sqrt{3} + \sqrt{5} \: is \: irrational$

Answer 1

let us assume that √3 is rational
therefore √3 = p/q where p and q are coprime integers
therefore,
√3 = p/q
√3q = p
3q² = p²
therefore p is divisible by 3
therefore,
p = 3r
p² = 9r²
3q² = 9r²
q² = 3r²
therefore q is also divisible by 3
but p and q are coprime
therefore contradiction arises
thus assumption is incorrect and √3 is irrational.

let us assume that √5 is rational
therefore √5 = a/b where a and b are coprime integers
therefore,
√5 = a/b
√5b = a
5b² = a²
therefore p is divisible by 5
therefore,
a = 5c
a² = 25c²
5b² = 25c²
b² = 5c²
therefore q is also divisible by 5
but p and q are coprime
therefore contradiction arises
thus assumption is incorrect and √5 is irrational.

√3+√5 = irrational + irrational
therefore = irrational
hence proved