## Why can we square both sides?

### June 8, 2010

This post is going to be a short post, and a deviation from what I’ve normally written about; I have been taking a slight break from linear algebra, but expect a post soon regarding vector spaces and lots of stuff we can do with them.

The title of this post comes from a student that I had recently, asking why “squaring both sides” of an equations is legitimate.

I had honestly never deeply thought about why it is the case that if $a = b$ then $a^n = b^n$ for any positive integer power $n$, at least while I’ve been tutoring high school maths.  One particularly telling algorithmic proof that is relatively easy and helpful to go through with the student is:

Given that $a = b$, then we multiply both sides of the equation by $a$.  We then have $a\cdot a = b\cdot a$.  But since $a = b$, replace $a$ on the right side to give us $a\cdot a = b\cdot b$.  This is obviously equivalent to $a^2 = b^2$, and to get any other power, just continue.

If the student is ambitious, then one can even extend it to negative powers of both sides.  But what about fractional powers?  Irrational powers?  Hm.  Ask the student what they think.

Incidentally, what do my readers think of showing proofs to high school students who don’t necessarily like mathematics?  My rule for students is: if they do not know the formula, they must derive it.  For example, if the student does not remember the sides of the 45-45-90 triangle, they must re-derive it.  I force my students to prove the formula for the sum of angles in a polygon.  I think that it helps them understand where the formula comes from, but I’ve talked to other teachers that say kids should do the “how” first, and leave the “why” to kids who do mathematics for a living.

edit: As brk so graciously pointed out, the problem above is not asked anywhere, because it’s exceedingly trivial: if $a = b$, then it is always the case that $a^n = b^n$ for any n, simply by substitution.  I may have been thinking about continuity or something silly like that.  Either way, the point of this post was to illustrate a nice way of helping students prove to themselves that squaring both sides is a “legit” thing to do.  The fact that $a = b$ and the substitution proof is lost on a few students (especially in things like $\sqrt{x^2 + 2x}= 4$), so I wanted to just mention this algorithm-proof fo’ any interested teachers ’cause I think it’s kind of cute.