[Math] Stuck proving that if $m$ and $n$ are perfect squares. Then $m+n+2\sqrt{mn}$ is also a perfect square.

Question

I am relatively new to proofs and can't seem to figure out how to solve an exercise.

I am trying to prove:

Suppose that $m$ and $n$ are perfect squares. Then $m+n+2\sqrt{mn}$ is also a perfect square.

I know that per the definition of a perfect square, that $m=a^2$ and $n=b^2$, if a and b are some positive integer.

I can then use substitution to rewrite the statement as:

$$a^2+b^2+2\sqrt{a^2b^2}$$

I also know that $2\sqrt{a^2b^2}$ can be simplified to:
S
$$a^2+b^2+2ab$$

I am stuck after this point though. I don't know how to eliminate the $2ab$.

Answer 1

You don't need to eliminate the $2ab$ term.

Notice that $(a+b)^2 = (a+b)(a+b) = a^2+ab+ba+b^2 = a^2+b^2+2ab$.