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
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