Equality Properties and What They Really Mean

Date: 07/30/2008 at 12:27:13
From: Margaret
Subject: Is there a property of equality for powers and roots?

In class we are shown how to square both sides of an equation or take
the square root of both sides of an equation but is there a rule like
the addition property of equality or multiplication property of
equality that says it is ok to do so?

I have asked the instructor, looked at algebra text books and searched
Dr. Math.

Date: 07/30/2008 at 16:06:30
From: Doctor Peterson
Subject: Re: Is there a property of equality for powers and roots?

Hi, Margaret.

Interesting question!  I think it shows that these properties really 
shouldn't be taught in this way, which makes things simple for 
teaching kids but doesn't accurately reflect what is actually 
happening.  The so-called addition and multiplication (and 
subtraction and division) properties of equality are not really 
properties of equality in the first place, but are facts about each 
operation.

I believe you are talking about facts like this, the multiplication 
property of equality:

  If a = b and c is any real number, then ac = bc.

The idea is that if two numbers are really the same number, then when 
we multiply them both by the same thing, we get the same answer.  How 
could we not?  As long as multiplication is "well-defined"--that is, 
always gives the same answer--this has to be true.  The same is true 
of any other operation, including powers, square roots, reciprocals, 
and so on!  Any well-defined operation (or function, in fact) will 
behave this way.  The only thing that could go wrong, really, is if 
you can't perform the operation at all (e.g. if you want to take the 
square root of both sides but one or both may be negative).  This 
becomes a domain issue, if you are familiar with functions.

So you don't really have to look for specific properties of equality 
associated with each operation you want to use; you just have to 
determine that it is well-defined (has one value) and that its domain 
includes the values to which it is being applied.  These two facts 
amount to the property you are looking for.

Now, some texts define the "X property of equality" as something a 
bit deeper than what I have just discussed:

  If c is any real number other than 0, then the equation ac = bc
  is EQUIVALENT to the equation a = b.

This is what you REALLY need to use when you solve an equation; it 
says not only that ac is still equal to bc, but that they are ONLY 
equal if a = b; you don't either lose or gain solutions.

This property is not necessarily true for any well-defined operation 
(or for any function), but for any INVERTIBLE (one-to-one) function 
(that is, when there is only one way to get any given result).  In 
particular, it is NOT true for even powers, because there are two 
different numbers with the same square, so that for example 1 = -1 is 
not true yet (1)^2 = (-1)^2 is true.  They are not equivalent.  This
is why squaring both sides of an equation can yield extraneous solutions.

A related issue comes up with square roots.  Although it is true that 
if a = b, then sqrt(a) = sqrt(b), and in fact these equations are 
equivalent if you ignore domain issues, this can lead to problems 
when you forget that the radical symbol means only the positive root, 
and try to apply it to something like this:

  (-1)^2 = 1^2

Taking the square root of each side by just canceling out the 
squares, you'd get

  -1 = 1

which is not true!  You might not do this here, but you probably 
would if there were variables:

  x^2 = y^2

does not imply that

  x = y

because one might be positive and the other negative!  What's 
happening here is that, on one hand, you are unconsciously using a 
form of the square root that is not a function (that is, has more 
than one value) by allowing a negative result; or, on the other hand, 
you are forgetting that in reality

  sqrt(x^2) = |x|, not x.

So the answer to your question really depends on exactly how your 
"multiplication property" and so on are defined, and what you want to 
use your new property to do.  Perhaps if you show me how you want to 
use it, I can clarify what I am saying.  Hopefully in taking squares 
or square roots in equations you have been taught the caveats that 
arise; some books may present these facts as properties, including 
all the warnings, but others pass by them all too quietly!

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 07/31/2008 at 15:44:32
From: Margaret
Subject: Thank you (Is there a property of equality for powers and roots?)

Thank you for your help. I need to think about these very interesting
ideas and look forward to sharing them with my instructor when we meet
for follow up in two weeks.

Margaret