Squares Squared

Let’s say we’re going to calculate the product, P, of two integers, X and Y, each of which are squares.  Their product will also be a square:

P = X • Y

But since X and Y are squares, they can each be broken down into their roots, X_r and Yr, and the problem rewritten as follows:

P = X_r • X_r • Y_r • Y_r

Next, we can rearrange the terms like so:

P = X_r • Y_r • X_r • Y_r

P = (X_r • Y_r) • (X_r • Y_r)

P = (X_r • Y_r)²

Therefore, P will be a square.