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:


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 $2ab$.


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

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

  • $\begingroup$ Thanks for the detail in your answer! I was definitely not seeing the connection until you walked me through it! $\endgroup$
    – timbram
    Jan 17 '16 at 4:19

Now use the fact that $a^2+b^2+2ab = (a+b)^2$.


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