Simplifying Exponential Equations

Date: 08/19/2003 at 17:51:41
From: Mark
Subject: Simplifying exponential equations

I have always had trouble with exponents, and this is a review 
question that I have no clue how to even start. It is (3r^-1s^2t^3)
^2/27r^3st^-2. I don't even know how to begin! Can you help?

Date: 08/20/2003 at 13:46:30
From: Doctor Ian
Subject: Re: Simplifying exponential equations

Hi Mark, 

You can find a simple explanation of the important rules here:

   Properties of Exponents
   http://mathforum.org/library/drmath/view/57293.html 

Be sure to read it before continuing. 

Given your example, 

  (3 r^-1 s^2 t^3)^2
  ------------------
    27 r^3 s t^-2

the easiest rule to apply would be 

   (ab)^c = (a)^c (b)^c

We can use this to rewrite the numerator:

  (3)^2  (r^-1)^2 (s^2)^2 (t^3)^2
  -------------------------------
         27 r^3 s t^-2

Now the easiest rule to apply is 

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

That gives us 

   9  r^(-2)  s^4  t^6
  ---------------------
  27  r^3     s    t^-2

Now, at this point, it helps to keep things straight if we
'unmultiply' the fraction:

   9    r^(-2)   s^4   t^6
  -- * ------- * --- * ----
  27    r^3      s     t^-2

Why would we do this?  Because now we can apply the rule

   a^b
   --- = a^(b-c)
   a^c

For example, 

  t^6
  ---- = t^(6 - -2) = t^(6 + 2) = t^8
  t^-2

So now we have

   9    r^(-2)   s^4   
  -- * ------- * --- * t^8
  27    r^3      s     

I'll leave the rest to you, but write back if you're still stuck. 
 
As for the logic of these problems, it's basically the same as the
logic of most of mathematics: You start out with something that
doesn't look the way you want it to, and you apply successive
transformations to it. If each transformation is guaranteed to
preserve the meaning of the thing, then you can change appearance
without changing meaning.  

For example, this is all you're really doing when you evaluate an
arithmetic expression, 

     3 * (4 - 2) + 6 / 3

   = 3 * 2 + 6 / 3               Because 4 - 2 = 2

   = 6 + 6 / 3                   Because 3 * 2 = 6

   = 6 + 2                       Because 6 / 3 = 2

   = 8                           Because 6 + 2 = 8

or when you substitute a value into an equation, 

    y = 3x - 4

    y = 3*2 - 4                  Legal if we know that x = 2
 
    y = 6 - 4                    Because 3 * 2 = 6

    y = 2                        Because 6 - 4 = 2

or when you solve a formula for one variable in terms of others, 

           V = pi r^2 h          Volume of cylinder

        V/pi = r^2 h             Dividing both sides by the 
                                 same (nonzero) quantity preserves
                                 equality. 

      V/pi/h = r^2               Same thing. 

     _______     ____
   \| V/pi/h = \| r^2            Taking the square root of both
                                 sides preserves equality.

     _______                               ____________
   \| V/pi/h = r                 Because \| something^2  = something


or when you integrate, or differentiate, or prove a trig identity, and
so on, and so on. 

It's really just the same game, played in slightly different contexts,
i.e., you have rules that say "Whenever I see an instance of THIS, I
can replace it with an instance of THAT," and then you search for
opportunities to apply these rules.  

How do you know which rules to apply at any given time? That's sort
of like knowing which piece to move next in a game of chess. Largely
it's a case of knowing what to do in a particular situation because
you've been in that situation before.  

I hope this helps!

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