Fractions with Exponents

Date: 01/29/2001 at 16:57:41
From: dawn
Subject: Fractions with exponents

I need help with questions like (-3x^2 y^2)^-2/(xyz)^-2. I do not 
understand why the answers are what they turn out to be.

Date: 01/29/2001 at 17:43:52
From: Doctor Ian
Subject: Re: Fractions with exponents

Hi Dawn,

The thing you need to know to deal with problems like these is that

          1
  a^-b = ---
         a^b

So when you see a fraction like 

  (-4x^3 * y^5 * z^-2)^-2 
  -----------------------
         (xyz)^-3

the first thing to do is to take everything with a negative exponent 
and move it from the numerator to the denominator, or from the 
denominator to the numerator, changing the sign of the exponent as 
you go:

           (xyz)^3
  = ----------------------
    (-4x^3 * y^5 * z^-2)^2

Note that I didn't change z^-2 yet, because it's inside a pair of
parentheses.  

The next thing you need to know is that

  (a^b)^c = a^(bc)

So we can use that fact to get rid of the parentheses:

            x^3 * y^3 * z^3
  = ------------------------------
    -4x^(3*2) * y^(5*2) * z^(-2*2)


      x^3 * y^3 * z^3
  = --------------------
     -4x^6 * y^10 * z^-4

Now we can use our first rule again:

      x^3 * y^3 * z^3 * z^4
  = -----------------------
         -4x^6 * y^10 


You also need to know that 

  a^b * a^c = a^(b+c)


So we can combine the z's on top:

      x^3 * y^3 * z^7
  = -----------------------
         -4x^6 * y^10 

Now all that's left to do is cancel things out:

          z^7
  = --------------
      -4x^3 * y^7

If you were able to follow all this, then you should be able to use 
the same techniques to solve your own problem.  

If you weren't able to follow all this, please write back and tell me 
which steps I didn't explain completely enough, and I'll try again. 

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