Combining Positive and Negative Exponents

Date: 06/30/99 at 12:33:38
From: Lauren Fortner
Subject: Algebra: using the power theorem

I understand how to get problems from "beginning" form, for example:

     x (x^-3)^2 y (xy^-2)^-3
     -----------------------
      (y^2)^3 y^-3 (x^2)^3

to "after" you use power theorem:

     x x^-6 y x^-3 y^6
     -----------------
       y^6 y^-3 x^6

and then when you simplify the numerator and denominator:

     x^-8 y^7
     --------
     y^3 x^6

What I don't understand is how you get from that to writing all 
exponential expressions with positive exponents, or negative 
exponents, or with both. I homeschool, so I don't really have anyone 
to ask.

Thank you for your help.

Date: 07/01/99 at 17:00:53
From: Doctor Peterson
Subject: Re: Algebra: using the power theorem

Hi, Lauren. Thanks for a well-written question - it really helps to 
know just what part you have trouble with.

The key to this step is that

     x^-a    1           1     x^a
     ---- = ---   and   ---- = ---
      1     x^a         x^-a    1

That's essentially just the definition of negative exponents, and I'll 
assume you're at least aware of this as a fact. What you need is how 
to apply it.

What these facts mean in practice is that you can move a factor from 
top to bottom or from bottom to top and negate the exponent. Once you 
get used to it, you just think "There's an x^-8 in the numerator, so I 
can replace that with an x^8 in the denominator." To take it more 
slowly, we can pull the expression apart, apply the rule, and put it 
back together:

     x^-8 y^7   x^-8   y^7    1     1     1    y^7    1     1
     -------- = ---- * --- * --- * --- = --- * --- * --- * ---
     y^3 x^6      1     1    y^3   x^6   x^8    1    y^3   x^6

                    y^7
              = -----------
                x^8 y^3 x^6

That makes all the exponents positive, but there's still another step 
you didn't mention: combining like factors so that x and y each appear 
only once. That's easy in the denominator now; we can just permute so 
the x's are together and add the exponents:

         y^7         y^7
     ----------- = --------
     x^8 y^3 x^6   x^14 y^3

But you still have y in two places. We can use the same rule to get 
the y's together; since 7 > 3, let's move the y^3 to the top to keep 
the exponents positive:

       y^7      y^7    1      1    y^7    1     y^-3   y^7 y^-3   y^4
     -------- = --- * ---- * --- = --- * ---- * ---- = -------- = ----
     x^14 y^3    1    x^14   y^3    1    x^14     1      x^14     x^14

Now we're really done.

But we've taken a lot more steps than we had to. That's fine when 
you're starting out; this isn't a race. But here's how I'd do it 
myself:

     x(x^-3)^2y(xy^-2)^-3   x x^-6 y x^-3 y^6
     -------------------- = -----------------
     (y^2)^3y^-3(x^2)^3     y^6 y^-3 x^6

                          = x x^-6 y x^-3 y^6 * y^-6 y^3 x^-6

                            (Here I've moved everything to the top.)

                          = x x^-6 x^-3 x^-6 * y y^6 y^-6 y^3

                            (Here I've gathered the x's and y's 
                             together; I'd probably mark each factor 
                             as I copied it to make sure I didn't miss 
                             any.)

                          = x^(1-6-3-6) y^(1+6-6+3)

                          = x^-14 y^4

                            y^4
                          = ----
                            x^14

If you're at all afraid of negative exponents, this will make you 
dizzy, but if you like to take the bull by the horns and get it over 
with, making the negative exponents work for you, this is the way to 
do it. Basically, I decided I'd rather deal with positive and negative 
exponents than with numerators and denominators; they're really two 
ways to say the same thing, and mixing them doesn't make sense. So 
even though the goal is to have numerator and denominator and 
eliminate negative exponents, I found that it's really easier to work 
with the negatives, then change them to denominators when I'm done.

If any of this overwhelmed you, or if you have more questions, feel 
free to write back.

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