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Explaining Exponents

Date: 10/23/97 at 15:33:07
From: Amanda Glass
Subject: Exponents

I am Amanda and I do not understand how to solve exponents.
All I know is you use multiplication, and that is all I know
about exponents. Well, I guess I do know, for example, like 5 to the 
second power or like 6 to the third power. I was wondering if you 
would write back explaining to me how to do them. 

Thanks. Bye.

Date: 11/23/97 at 18:14:11
From: Doctor Mark
Subject: Re: Exponents

Hi Amanda,

If you know how to find 5 to the second power or 6 to the third power, 
then you already know more than most people. But I'm guessing that you 
aren't real sure about that, so let me describe what I know about 
exponents. If everything I say is stuff that you already know, then 
please write back to me and tell me what else you want to know, okay?

As you probably know, an exponent is a number that is written a little 
to the right, and a little above, another number (which is called the 
"base"). The exponent is usually written a little smaller than the 
other number (the base), but not necessarily. I can't write things 
like that in this message because my computer won't let me do it.  
What I can do is write this:  a^n. The funny looking "^" is called a 
"caret" or "hat" (which it sort of looks like, don't you think? - a 
hat, I mean) and it means that the "n" (the exponent) sits a little on 
top of the a, like a hat would. Here's what that little number (n) 
sitting on top of the hat means.

When you write a^n, it means this:

Write "n" of those "a"s down, in a line, leaving a space between them.
Then put a "times" symbol, like "x," between them, and multiply all 
those numbers together.

Here are some examples:

  3^2. The "2" is the exponent. This means:

    Write two (the exponent) of the 3's down:  3   3.  Now put an "x" 
    between them,  3 x 3, and multiply:

    3^2 = 3 x 3 = 9.

  2^3. The "3" is the exponent. This means:

    Write three of the 2's down: 2   2   2.  Now put an "x" between 
    them, 2 x 2 x 2, and multiply:

    2^3 = 2 x 2 x 2 = 8.

  7^4. The "4" is the exponent. This means:

    Write four of the 7's down: 7   7   7   7 .  Now put an "x" 
    between them, 7 x 7 x 7 x 7, and multiply:

    7^4 =7 x 7 x 7 x 7 = 2401 (you might have needed a calculator for

  4^8. The "8" is the exponent. This means:

    Write eight of the 4's down: 4   4   4   4   4   4   4   4 
    (when the exponent is big, you have to be careful to put down the 
    right number of 4's.  I just counted, and there are, in fact, 
    eight of them).  Now put an "x" between them, 
    4 x 4 x 4 x 4 x 4 x 4 x 4 x 4, and multiply:

    4^8 = 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 = Use your calculator!

(According to my calculator, that's equal to 65,536. Did you get that?  
If you didn't, maybe you didn't multiply the right number of 4's 
together. Notice that the exponent - here, 8 - is the number of times 
you push the "4" button on your calculator.)

Okay, now here's one that's a little tricky.  What is 5^1?

Follow the same rule. To find 5^1, we write one of the 5's down: 
That's pretty easy:


Okay, now all we have to do is multiply all those 5's together, so we 
better get out our calculator... But Amanda, we don't need a 
calculator to do that, do we?  There's only one 5 there, so it's the 
easiest thing in the world to multiply "all" those 5's, since there's 
nothing to multiply!

  5^1 = 5, and we're ***done***.

Maybe you can see that when the exponent is 1, it's really easy to 
find the answer: just write down the number underneath the exponent 
(well, here, it's the one on the left of that "hat"!).


  3^1 = 3,  73^1 = 73,  513^1 = 513, and so on.

Let me explain one kind of mistake that some people sometimes make.  
When they see 5^2, they think they get 10. They think 5 times 5 is 10.  
Do you know what they did wrong?  They thought that when you want to 
find 5^2, you write down two of the 5's and then write a ***plus*** 
(+) sign in between them (add them):

  5 + 5, which *is* 10.

But as we know, that's not 5^2! 5^2  means to write down two of the 
5's and write a ***x*** in between them, i.e., multiply them:

  5^2 = 5 x 5 = 25.

Of course, if I'm tired, or not concentrating, I might do something 
silly like say that 5^2 = 10, but not if I'm thinking about it.

I should say, Amanda, that the way I explained exponents here only 
works when the exponent is a "positive integer" [a "counting number"]  
(like 1, 2, 3, 7, or 26). It turns out that an exponent can be a 
negative number like - 4, or it could be 0, or it could even be 
something really wierd like 3.14159. If the exponent looks like that, 
it turns out that a^b still means something, but it means something a 
little different than what I said (of course: if you wanted to find 
out what 5 with an exponent of - 4 was, you would have to write 
"negative 4" of those 5's, and even I don't know how to do that!).  
You probably won't see exponents like that for another year or two, 
though, so don't worry about them now. When you do see them (or if 
you've already seen them!), you could write me back and I'll explain 
it to you.

I hope this is of help Amanda, and if  it isn't, make sure you write 
me back, okay?

-Doctor Mark,  The Math Forum
 Check out our web site!   
Associated Topics:
Middle School Exponents

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