(New Question of the Week)
I like working with a student who is willing to take chances, and also willing to be corrected. As I have often explained, just like a medical doctor, a Math Doctor wants you to “show me where it hurts” in order to diagnose the problem; so showing detailed work is valuable to us, especially when it is wrong!
Here is a good discussion earlier this month (with an old friend, Sarah, from the Ask Dr. Math site) that brought out some ideas that may be useful for many students. As usual, I am lightly editing the conversation (removing some typos on both sides), but leaving in significant mistakes where corrections were needed, because they can be very illuminating.
The question: expanding a difference of squares
Expand:
(3b^1/2 – 2b^-1/2) (3b^1/2 + 2b^-1/2)
I opened it up and got
9b + 6 – 6 – 4b^-1 = 9b – 4b^-1
However, since it is a difference of two squares, why doesn’t the answer become 9b^1/4 – 4b^1/4 ?
It sounds like I’m making a stupid mistake but when I asked my teacher, she couldn’t figure it out either.
If you are not familiar with the notation used in typing exponents without using superscripts (which were not available on our old site, and are awkward to use anyway), we write x^n to mean that the n is raised, xn.
Sarah has to “expand” or “open up” a product (which in other places might be called “multiplying out” or “distributing” or “FOILing”), and recognizes that it is related to the formula for factoring a difference of two squares,
\(x^2 – y^2 = (x – y)(x + y)\)
The two factors look like the factors on the right hand side of the formula, so she expects the result she got by distributing to match the left hand side. But her work in expanding the long way looked like this:
\((3b^{1/2} – 2b^{-1/2}) (3b^{1/2} + 2b^{-1/2}) = \) \((3b^{1/2})(3b^{1/2}) + (3b^{1/2})(2b^{-1/2}) – (2b^{-1/2})(3b^{1/2}) – (2b^{-1/2})(2b^{-1/2}) = \) \(9b + 6 – 6 – 4b^{-1} = 9b – 4b^{-1}\)
while the difference of squares (she thought) was
\((3b^{1/2})^2 – (2b^{-1/2})^2 = 9b^{1/4} – 4b^{-1/4}\)
Do you see her mistake? I did; but I wanted her to develop the skill to see it for herself:
Welcome to the new site, Sarah!
Can you tell me in more detail why you think it should be 9b^{1/4} – 4b^{1/4}? I have a suspicion what you are probably thinking, but it will be best for you if you show me your steps and perhaps discover the error for yourself.
It may also help if you show me the details on how you got the correct answer by expanding the product, for comparison.
(Yes, it can be called a “stupid” mistake, but those can be very much worth discussing!)
Sarah responded:
I got the correct answer simply by expanding the brackets, so
3b^0.5 * 3b^0.5 = 9b
3b^0.5 * 2b^-0.5 = 6b^0 = 6
-2b^-0.5 * 3b^0.5 = -6b^0 = -6
-2b^-0.5 * 2b^-0.5 = -4b^-1
Hence the answer 9b – 4b^-1.
As for the incorrect answer, I thought
(X^2 – Y^2) = (X – Y) (X + Y) = X^2 + XY – XY – Y^2 = (X^2 – Y^2)
So in the same way, (3b^1/2 – 2b^-1/2) (3b^1/2 + 2b^-1/2) = 9b^0.25 – 4b^0.25.
The 0.25 is 0.5^2. This part doesn’t feel right. In multiplication, you add the indices.
Thanks for not telling me immediately by the way. It’s actually better 🙂
(For our American readers, “indices” is the British way to say “exponents”.)
How to catch an error
This showed that she did the right work when simply expanding the product, but made the mistake I had expected when doing it the other way. I had hoped that writing out both in detail would make the error visible to her; but although the first way she correctly got \(b^{0.5} \cdot b^{0.5} = b^1\), adding the exponents, she didn’t explicitly show the step in the second method where she seems to have thought that \(\left(b^{0.5}\right)^2 = b^{0.25}\) (squaring the exponent rather than multiplying by 2), even though this means exactly the same thing. So much for my hopes! (Note that she said her teacher also didn’t see the problem; this is a human foible, not stupidity.)
I tried again:
You said,
I got the correct answer simply by expanding the brackets, so
3b^0.5 * 3b^0.5 = 9b
3b^0.5 * 2b^-0.5 = 6b^0 = 6
-2b^-0.5 * 3b^0.5 = – 6b^0 = -6
-2b^-0.5 * 2b^-0.5 = -4b^-1
Hence the answer 9b – 4b^-1.
Very good. Then,
As for the incorrect answer, I thought
(X^2 – Y^2) = (X – Y) (X + Y) = X^2 + XY – XY – Y^2 = (X^2 – Y^2)
So in the same way, (3b^1/2 – 2b^-1/2) (3b^1/2 + 2b^-1/2) = 9b^0.25 – 4b^0.25.
The 0.25 is 0.5^2.
Yes, we can directly multiply the sum and difference of the same two terms by writing the difference of the squares. But you didn’t quite write out enough details to see your error. Here is what you are claiming:
(3b^{1/2})^2 – (2b^{-1/2})^2 = 9b^{1/4} – 4b^{1/4}
Do you see what you did wrong in squaring each term? It’s a very common error, which is why I want to eradicate it by making sure you see it clearly!
Note, by the way, that I have added braces (which could just as well have been parentheses) around the fractional exponents, to make sure it is clear what is in the exponent. I could also have written your work as
(3b1/2)2 – (2b-1/2)2 = 9b1/4 – 4b1/4
Can you see the error now? I like to write out a little more than I think I need to; I tell my students that I am writing what I am going to do before I do it, so I can then look back and see whether I really did what I intended to do. This helps prevent those “stupid” mistakes that we all make when we think too fast!
Because we don’t yet have the best facilities for writing mathematical expressions, we still have to give guidance on how to write clearly; but when we eventually offer LaTeX formatting as an option, we’ll have to offer help with that, too. Part of helping students is helping with details of communication, even in math.
The best of us (and Sarah has made it clear previously that she is a very good student) can be temporarily blind to a mistake we are making. Sarah showed that she still didn’t see the error:
I do understand your point, but what I don’t understand is this:
X4 – Y 4 = (X2 – Y2)(X2 + Y2) because the square root of 4 is 2, so if you have 9b1/4 – 4b1/4 won’t it become (3b1/2 – 2b1/2)(3b1/2 + 2b1/2)?
Thanks.
The error: powers of powers
This is a very subtle error, easy to miss, especially if you see too often an example like \(\left(x^2\right)^2 = x^4\), which would give the same result whether you did \(2 + 2 = 4\), \(2 \cdot 2 = 4\), or \(2^2 = 4\). Sarah was walking right past the error, even pointing out what she was doing, without seeing that it was wrong. So I had to be direct at last:
Ah! But the point in your example is not that the square root of 4 is 2, but that half of 4 is 2!
The rule is that (a + b)(a – b) = a2 – b2. But when we square a power, we don’t square the exponent, we double it. For example, (x3)2 = x3*2 = x6, not x3^2 = x9.
So here is the correct work:
(3b1/2)2 – (2b-1/2)2 = 32 b(1/2)2 – 22 b(-1/2)2 = 9 b1 – 4 b-1 = 9 b – 4 b-1
Again, in squaring (3b1/2), we don’t square the exponent, 1/2, but rather multiply it by 2, which gives 1, not 1/4.
To put it another way, in what you wrote above, when you expand (3b1/2 – 2b1/2)(3b1/2 + 2b1/2), you would not multiply the 1/2’s, but add them, to get 9b – 4b.
Do you see now?
This time it worked:
Thank you! Makes sense! So just to confirm,
9b1/4 – 4b1/4 gives you (3b1/8 – 2b1/8) (3b1/8 + 2b1/8)
Your explanations are really clear. Thanks for showing me my mistake 🙂
That was a nice way to check, by taking her result and doing the factoring, showing that she doesn’t get what she started with, and confirming her understanding.
That’s right. When you use this formula as a factoring method, you obtain each term by taking the square root of the coefficient, and half the exponent. I like to write the intermediate step like this:
9b1/4 – 4b1/4 = (3b1/8)2 – (2b1/8)2 = (3b1/8 – 2b1/8)(3b1/8 + 2b1/8)
Ultimately, there were two lessons I was setting out to teach: how to square a power, and how to avoid missing the fact that you are squaring a power. Knowing facts is important, but developing habits that help you always apply those facts when they are appropriate is perhaps even more important. It’s the difference between knowing what foods are good for you, and developing the habit of eating them! And that habit can be supported by making it visible (keeping the good foods at eye level?).
I’ve said it before, and I will say it again: Unless you write something, you don’t know that it is right. When it matters, you should always write more steps than you think you need to, because steps done “in your head” are not visible, which makes them hard to check. If you don’t see what you did, you won’t notice that it is wrong.