Exponents in the Real World

Date: 11/23/97 at 08:36:35
From: Cairna
Subject: Exponents

How come we have to know how to add, subtract, multiply, and divide 
exponents?

In the real world when do you use exponents?  When do you add them? 
When do you subtract them? What about multiplying exponents? When do 
you divide exponents?

Do teachers just teach us them because they can?
I HAVE TO KNOW!

Date: 11/25/97 at 14:47:02
From: Doctor Mark
Subject: Re: Exponents

Hi Cairna,

Well, you asked the right guy, because I KNOW.

First, there's really no reason at all that you *have* to use 
exponents. If you've got loads of free time, you could just write out

5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5

instead of 5^18 (and if you've got a calculator, you could even find 
out what it's equal to!).  (The little "^" symbol is called a "hat" or 
"caret"; my computer won't let me write exponents in the normal way, 
so I have to use the ^ to mean that the number following it is the 
exponent.)

In fact, until about 400 years ago, nobody used exponents, and they 
were perfectly able to do mathematics. But then some people decided 
that it got boring always writing out all those "times" symbols, and 
they decided to use exponents to mean how many times you wrote down 
the number.

I hope you know that when a number is written with an exponent, the 
whole shebang is called an exponentIAL. So 5^18 is an exponential, 
18 is the exponent (also sometimes called the "power"), and the 5 is 
called the "base."  

There are rules for exponents only when the bases are the *same*, so 
while there is a rule for multiplying one power of 3 by another power 
of 3, there isn't any simple rule for multiplying a power of 4 by, 
say, a power of 17.  So in everything I am going to say, you should 
assume that all of the bases are the same.

When you multiply exponentIALs, you add the exponents.  That actually 
makes sense:

   (2^3) x (2^5) = (2 x 2 x 2) x (2 x 2 x 2 x 2 x 2)

But look what you did: you wrote down 2 three times (that's what 2^3
means), then you wrote it down five times (that's what 2^5 means), so 
how many times did you write it down? DUH! You must have written it 
down 3 + 5 = 8 times. But (after we put in the "times" symbols) that's 
what we mean by 2^8.  So,

   (2^3) x (2^5) = 2^(3+5) = 2^8.

That is, when you multiply exponentIALs, you ADD the exponents.

When you divide exponentIALs, look what happens:

   (2^7)/(2^4) = (2 x 2 x 2 x 2 x 2 x 2 x 2)/(2 x 2 x 2 x 2)

(this will look better if you write it out as a fraction. My computer 
won't let me do that, unfortunately). Now each of the 2s on the bottom 
cancels off one of the 2's on the top. So how many are left?

Let's see, we started out with seven of them on the top, and each of 
the four 2s on the bottom killed off one of the ones on the top, so on 
top, we have seven take away four - that's three - yes, we must have 
three left on the top, so that

   (2^7)/(2^4)= 2 x 2 x 2 = 2^3.

Do you see what happened? Each 2 on the bottom killed off one of the 
2s on the top, so there were seven take away four left. But that's 
just like subtracting: 7 take away 4 is 7 - 4 = 3.

And that's the rule for dividing exponentIALs: subtract the exponent 
of the one on the bottom from the exponent of the one on the top.  
I remember that by thinking of the fraction bar as a big minus sign 
for the exponents, so that

   5^8
  -----  <-----like a big minus sign for the exponents
   5^6

gives 5^(8-6) = 5^2.

What about if you have a power of a power, like (4^2)^3 (this really 
looks better if you write it out!)

Well, that's pretty easy:

We know how to take the 3 power of anything: write it down 3 times, 
then multiply what you wrote down. So (4^2)^3 must mean to write down 
(4^2) three times, then multiply:

   (4^2)^3 = (4^2) x (4^2) x (4^2).

Now if we were to write out what 4^2 meant, we would write down two of 
the 4s, then two more of the 4s, then two more of the 4s, and then 
multiply them all together. But how many 4s did we write down? You 
could count them, of course, but notice that we wrote down two 4s, and 
we did that three times. So we must have written down two times three 
of the 4s, i.e., six of the 4s. That is,

   (4^2)^3 = 4^(2x3) = 4^6.

This then gives us the rule for taking a power of a power: multiply 
the powers.

So, for instance, (7^4)^5 = 7^(4x5) = 7^20.

So we know when to add exponents, when to subtract them, and when to
multiply them. What's left? Oh yes, division. When do we *divide*
exponents?

Surprisingly, *never*.  About 5 or 6 years from now, when you take 
Algebra II, you will see something that looks sort of like dividing 
exponents, but it really won't be that: it will look like division, 
but it will really be multiplication (it has something to do with 
square roots, if you know what those are).

So now on to your excellent question about when do you use exponents.

Exponents are used all over the place in science, where they often 
appear as powers of the number 10 (it's called "scientific notation").  
In business, people use exponents to describe how much money they make 
when they sell, say, sports watches. And exponents were involved if 
your parents bought a house, or took out a loan to buy a car. They're 
used to describe acid rain, and to determine whether it's safe to go 
swimming in a swimming pool (if you have a swimming pool, you've 
probably heard your mother or father talk about the "pH" level of the 
water--and it *doesn't* mean what you think!), earthquakes (maybe 
you've heard of the Richter scale), how loud sound is (the "decibel" 
level), and how bright stars are (if you or any of your friends are 
interested in astronomy, you might have heard of the "magnitude" of a 
star or galaxy). All these things are described using exponents, and 
the laws of exponents are used to determine, say, how much stronger 
one earthquake is than another.

I hope this has been of help. Write back if you have any other 
questions.

-Doctor Mark,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/