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, Xr and Yr, and the problem rewritten as follows:

P = Xr • Xr • Yr • Yr

Next, we can rearrange the terms like so:

P = Xr • Yr • Xr • Yr

P = (Xr • Yr) • (Xr • Yr)

P = (Xr • Yr)2

Therefore, P will be a square.