Say, $a=b+c$, $a$ may be rational or irrational. However, the constraint on $a$ is that $a^2$ is an integer. b>0 , c>0 which means a>0. Wanted to confirm that either $b^2$ or $c^2$ or both can't be irrational.
I reasoned like this : $a^2 =(b+c)^2$ or $a^2=b^2+2bc+c^2\tag{1}$ Since the sum of a rational and irrational number is irrational (is it correct?), and the left hand side of $(1)$ is $a^2$ which is an integer, i.e., a rational number, neither $b^2$ nor $c^2$ can be irrational. For that matter, $2bc$ can't be irrational either. Conclusion: $a^2, b^2$ and $bc$ are rational numbers.
If any of my friends in this forum can help me confirm that above conclusion is correct, I will be grateful.
Off the top of my head, I could think of b = 2^(1/4) and c = -2^(1/4) as counterexamples.