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Negative Exponents: Who Needs Them?

Date: 09/19/2011 at 13:57:44
From: George
Subject: Negative Exponents

Why would someone use negative exponents?



Date: 09/20/2011 at 20:53:00
From: Doctor Peterson
Subject: Re: Negative Exponents

Hi, George.

You may have noticed that after a text introduces negative exponents, it
says that you will not be allowed to use them in answers to exercises,
because that would not be considered to be "simplified."

So why bother?

Let's see what uses I can find for them.

First, they are very useful in intermediate steps. When you have an
expression like ...

   x^4/x^7,

... you can first write it more compactly, as ...

   x^-3, 

... and then rewrite it in your final answer as need be. When the
expression is more complicated, this can be a very good strategy, because
you don't have to worry about which power goes where. You just follow the
same rule all the time.

There are some formulas in which using a negative exponent keeps them from
looking ugly. For example, check out our Frequently Asked Question about
loans and interest:

    http://mathforum.org/dr.math/faq/faq.interest.html 

In this FAQ's section on Compound Interest, we write

  Then the present value is given by P = A(1 + [i/q])^(-nq).

Without negative exponents, we would have had to have written this as

                                                A
  Then the present value is given by P = ----------------.
                                         (1 + [i/q])^(nq)

This notation becomes more useful later, under Installment Loans, when we
write

   The amount of the fixed payment is determined by

      M = Pi/[q(1 - [1 + (i/q)]^[-nq])].

Without the negative exponent, this would be

      M = Pi/[q(1 - 1/[1 + (i/q)]^[nq])].

Can you even see what this means? Written out more clearly,

                   Pi
      M = -----------------------
                       1
          q(1 - ----------------)
                [1 + (i/q)]^[nq]

If nothing else, using the negative exponent here makes it easier to put
into a calculator!

In science, you may see units written using negative exponents; this makes
it easier to write complicated units all on one line. For example, a speed
in meters per second (m/s) can be written as m.s^-1. (In that expression,
"." represents a raised dot for multiplication.) Again, this doesn't look
useful in such a simple case, but when units pile up, like J/(A.s),
writing J.A^-1.s^-1 helps to keep everything sorted out.

Probably the most important reason for negative exponents arises when an
exponent is a variable rather than just a number. You may not have seen
such exponential functions yet, but you probably will soon. When we write,
say, 10^x, x may be positive or negative, so we have to know what it means
in the latter case. There's no way to avoid it!

So the answer is actually a lot like the answer to "Why do we need
negative numbers?" We could get away without them in many cases where we
already know the sign; for example, instead of  ...

   5 + -3,

... we can write 

   5 - 3. 

And instead of ...

   3 - 5 = -2,

... we could say 

   "I owe you 5 - 3 = 2 dollars."

But when we want a formula that will apply to ANY numbers, and when we
need a variable that can describe any location, left or right of 0, we
need negatives. And when we don't know whether an exponent will be
positive or negative, we need negative exponents.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
Middle School Exponents
Middle School Negative Numbers

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