Question

a, b are positive rational numbers.
prove that a^(1/2) + b^(1/2) is rational if and only if a^(1/2) is rational and b^(1/2) is rational.

the <-- is easy.

a^(1/2) = p/q
b^(1/2) = r/s

a^(1/2)+b^(1/2) = p/q + r/s = (ps+rq)/(qs)

can't figure out the -->

Answer 1

The square root of any integer other than a perfect square is irrational, the question is how to show that.

Assume a^(1/2) = r/s (r,s integers in lowest terms, a not a perfect square) and derive a contradiction.