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### Exponents in the Real World

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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/
```
Associated Topics:
Middle School Exponents

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