How to do a Problem with Multiple Exponents

Date: Sat, 21 Jan 1995 09:55:21 AST
From: Richard Seguin
Subject:

Could you explain How you do this type of question? 

(5a^2b^3)(-2a^4b^7)
--------------------
(ab^2)^2

Date: 22 Jan 1995 15:44:17 -0500
From: Phil Spector
Subject: Re:

Richard-

I'm a little unclear about what the above formula really is.  It pretty
much depends on where one does or doesn't put the parentheses.  There are a
number of ways one could interpret the formula you typed in.  I'm going to
assume it's the following, because it makes the most sense.  If I'm wrong,
write back and scream at me and I'll solve it a different way:

     2 3     4 7
  (5a b )(-2a b )
-------------------
            2
         (2)
       (ab)

The thing to note first of all is that you can distribute terms throughout
the numerator.  In fact, a good idea might be to distribute the terms so
that "like terms" are next to each other.  For example, so that the
numbers, the a's, and the b's are next to each other.  Then, the numerator
looks like:
                                2   4   3   7
                       (5)(-2)(a )(a )(b )(b )

When you multiply like terms with exponents, there's an easy rule.  Just
add the exponents.  Note, for example, that if you wanted to multiply 2^2
and 2^3, that it would equal 2^(2+3) = 2^5 = 32.  (2^2=4, 2^3=8 => 2^2 +
2^3 = (4)(8) = 32)
Similarly, (x^2)(x^3) = x^(2+3) = x^5.  Therefore, (a^2)(a^4) = a^(2+4) = a^6,
and a similar rule applies to the b's.  So finally, you get:
                                       6   10
                                (-10)(a )(b  )
                                                          2
Now, with the denominator, you're squaring the quantity ab .  This is the
same as squaring each of the terms within the parenthesis.  So you'll end
up with
                               2
                         2 ( 2)
                        a (b )

This looks pretty weird, of course.  How can you square something twice?
Actually, when you are taking a term with an exponent to a power, you
simply multiply the two powers together.  For example,
                            3                    3
                          (2 )   6             (2 )    3
                       ( 2 ) =  2  = 64  Or  (2 )  = (4 )  = 64.

So in the case of denominator, we end up with :

                         2 4
                        a b .
                                              6 10
We now have the following equation:       -10a b
                                        ------------
                                              2 4
                                             a b

To divide like terms with exponents, it's a lot like multiplying like terms
with exponents, except you subtract instead of add the exponents together.
Why do you think this is?  I'm going to let you try to finish the problem
from here.  Hope I've been of help, and write back if there are any more
problems...

Phil, Dr. Math