Question

if p,q are positive integer, prove that √p +√q is an irrational

Answer 1

Let √p and √q be irrational numbers .
We have to prove that the sum of two irrational numbers is irrational.
Let √p+√q = a/b,where a and b are co primes and let √p+√q be rational
Squaring on both sides
(√p+√q)^2=(a/b)^2
(√p)^2+2.(√p).(√q)+(√q)^2=a^2/b^2
p+2√pq+q=a^2/b^2
2√pq=a^2-p-q/b^2
√pq=a^2-p-q/2b^2

Since a^2-p-q/2b^2 is rational, then √pq is also rational.
Which contradicts the fact that √pq is rational
√pq is irrational
Therefore,√p + √q is irrational