Raising Sides of an Equation to the 0 Power

Date: 12/04/97 at 18:53:35
From: Mr. LaVergne's 6th Hour Algebra 2H class
Subject: Raising sides of an equation to the 0 power

If you raise both sides of an equation to the 0 power, will both 
sides then be equal to one? You could make three to 0 power equal to 
ten to the 0 power if this is true.  Please enlighten us.

Date: 12/16/97 at 12:06:21
From: Doctor Mark
Subject: Re: Raising sides of an equation to the 0 power

Hello, 2H

Sure...

The short answer to your question is, yes.

The longer answer to your question is....

This is not really any different from a number of other things you 
could do:

   You could multiply both sides of an equation by 0
   You could divide both sides of an equation by 0.

But then there are two questions I would ask you:

   1. Why would you bother?
   2. What would you do with the result?

Let me explain...

Suppose we take some equation, say a *false* one like 1 = 2.

Let's multiply both sides of this false equation by 0.  Then we get:

   1 x 0 = 2 x 0 ------->  0 = 0.

Well, that's true isn't it?  But so what?  The fact that we *got* a 
true equation when we multiplied both sides of the equation by 0 
doesn't mean that the original equation was true, right? I mean, we 
know that 1 is not equal to 2!

Okay, you say, but what if we divide both sides of the false equation 
1 = 2 by 0.  Well, you know what happens, right?  You can't divide by 
0, so our false equation becomes

   one meaningless quantity = another meaningless quantity.

What are we going to do with that?  It's not true, nor is it false, 
since it has no meaning.

So what happens if we take the 0 power of both sides of the false 
equation 1 = 2?  We get:

   1^0 = 2^0 ---------->  1 = 1.

Well, that's true, but again, so what?  The fact that if you get 
something true by doing something to the original equation doesn't 
make the original equation any truer than it was (or wasn't) 
originally.

So the real point here is this: You *can* multiply both sides by 0, or
divide both sides by 0, or take the 0 power of both sides, but the 
result of doing any of those things tells you *absolutely nothing* 
about the original equation. So why bother?

Aha, you say: but what if I take an equation like 3x = 6. If I divide 
both sides by 3, I get x = 2. If I want that to be true, x better be 
(ahem!) 2, so that tells me that the original equation must be true 
for x being 2 also. Why does the fact that x = 2 being true for x 
taking the value "2" tell me that x taking the value "2" also makes 
the *original* equation (3x = 6) true? Didn't I say earlier that when 
you multiply both sides by 0 to get 0 = 0 (the equation I *got*), 
which is a true equation, that didn't mean that the *original* 
equation was true? So why does x = 2 (the equation I *got* when I 
divided by 3) being true mean that the *original* equation (3x = 6) is 
true? Aren't the two situations the same?

No, they aren't. And here is why.

When you divide the original equation by 3 to get x = 2, that is a
*reversible* step: You could multiply both sides by 3 (the number you
divided by) to get 3x = 6, the original equation, so if x = 2 is true, 
then 3x = 6 is true.

When you multiply both sides by 0,to get 0 = 0, that is *not* a 
reversible step, since to reverse the step, you would have to *divide* 
both sides of the equation 0 = 0 by 0, and that's a Big No No. You 
don't get *anything*. In particular, you don't get 1 = 2!

Similarly, if you were to divide both sides of the equation by 0, that 
is also not a reversible step, since to reverse it, you would have to 
multiply both sides by 0. That seems like it might be okay, but it 
isn't, since you *don't have anything to multiply by 0*, because (in 
the equation you got after you divided both sides by 0) both sides are 
meaningless.

A similar argument holds for taking the 0 power of both sides of the
equation; it's not reversible, since to reverse it, you would have to 
take the 0th *root* of both sides of 1 = 1, and what is meant by 
"taking the 0th root"?

[Here's the analogy:  If you square both sides of an equation, you 
"undo" that by taking the square root: that's the basis of solving a 
quadratic equation by completing the square, for example. If you cube 
both sides of an equation, you undo that by taking the cube root of 
both sides. So if you take the 0 power of both sides of an equation, 
you would have to undo that by taking the 0th root.]

A really good question, 2H, and I am glad you asked it.  I hope what I 
have said helps...

-Doctor Mark,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/