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Properties of Exponents


Date: 01/22/2001 at 11:28:35
From: Crystal
Subject: Properties of exponents

I was out of school with the flu, and when I came back my class had 
studied the unit "Properties of exponents." Will you please help me 
understand exponents?

Thanks.


Date: 01/22/2001 at 14:10:37
From: Doctor Ian
Subject: Re: Properties of exponents

Hi Crystal, 

What I'm going to try to do is start from the definition of what an 
exponent is, and show how you can figure the properties out from the 
definition, if you happen to forget them during a test, or whenever. 

So, the first thing to know about exponents is that they are just a
shorthand notation for a special kind of multiplication.  That is, 
there are times when I need to refer to a number like 

  6 * 6 * 6 * 6 * 6

but I don't want to keep writing that down all the time. So we have a 
much nicer notation, which is that

                       5      
  6 * 6 * 6 * 6 * 6 = 6         How you write it with a pencil.

                    = 6^5       How you write it on a keyboard. 

So, whenever you see something that looks like

  a^b

you know that it means 'b copies of a, all multiplied together'.  Here 
are some more examples:

        2^3 = 2 * 2 * 2

        3^2 = 3 * 3

  (a + b)^4 = (a + b) * (a + b) * (a + b) * (a + b) 

    (4^5)^2 = (4^5) * (4^5)


It's just an abbreviation, like writing 'Dr.' instead of 'Doctor', or 
like writing 'MA' instead of 'Massachusetts'. You need to be very 
clear on that, or nothing else about exponents is going to make sense.  

I always forget when I have to add or multiply exponents, so I use 
little examples to help me remember.  For example, if I have

  (a * a) * (a * a * a)  

that's (a^2)*(a^3), and I know the answer has to be a^5 (just from 
counting the a's), so that tells me that

  a^b * a^c = a^(b+c)

This, then, is one of the 'properties of exponents'.  

Here is another one.  If I have something like 

  (a * a) * (a * a) * (a * a)

that's (a^2)^3, and I know the answer has to be a^6 (again, just from
counting the a's), so that tells me that

  (a^b)^c = a^(b*c)

Note that if I have something like

  a * a * a
  ---------
    a * a

that's a^3 / a^2, and I know the answer has to be a (which is what's 
left after I cancel out as many a's as I can), so that tells me that

  a^b / a^c = a^(b-c)

This, by the way, is how we can figure out what negative exponents 
mean. Let's flip that fraction upside-down:

    a * a
  ----------  
  a * a * a

This is a^2 / a^3, and I know the answer has to be 1/a; and I just 
figured out that a^2 / a^3 = a^(2-3) = a^(-1). So that tells me that

             1
  a^(-b) = -----
            a^b

And what about this?

  a * a 
  ----- = ?
  a * a

Well, this is a^2 / a^2, which must be a^(2-2) = a^0. But I know that 
it has to be 1, since anything divided by itself must be 1. So this 
tells me that

  a^0 = 1

for any value of a, which seems a little weird, but it follows from 
the definitions we've been working out, and it doesn't lead to any 
bizarre consequences, so we just accept it. 

You can do a little more with exponents, too.  Note that, according to 
the rules that we just figured out, 

  a^(1/2) * a^(1/2) = a^(1/2 + 1/2) = a^1 = a

So a^(1/2) multiplied by itself is a... which tells me that

              __
  a^(1/2) = \/ a

               ___
           \3 /
  a^(1/3) = \/  a


and so on.  

So far, we've been talking about different exponents with the same base. 
But consider a situation where you have the product of two bases raised
to the same exponent, e.g.,
     
  a^3 * b^3 = (a * a * a) * (b * b * b)

            = (a*b) * (a*b) * (a*b)

            = (a*b)^3

That is, we just pair up the bases.  In the general case, 

  a^c * b^c = (ab)^c

This is very useful for simplifying products of square roots, e.g., 

    __     ___
  \| 6 * \| 24  = 6^(1/2) * 24^(1/2) 

                = (6*24)^(1/2)

                = (144)^1/2
                    ____
                = \| 144

                = 12
       
To summarize, here are the interesting properties of exponents:

  a^c * b^c = (a*b)^c

  a^b * a^c = a^(b+c)

  a^b / a^c = a^(b-c)

    (a^b)^c = a^(b*c)

     a^(-b) = 1 / (a^b)

        a^1 = a

        a^0 = 1

    a^(1/n) = the nth root of a

The main thing you need to do, I think, is avoid the feeling that 
these are a bunch of arbitrary things that you need to memorize, like 
the names of the state capitals. As you can see, they are all just 
_consequences_ that follow from the definition of an exponent, which 
is just itself something that mathematicians invented so they could be 
lazy about writing things down. And that means that if you really 
understand the definition, you can make up little examples like the 
ones I've made up above to rediscover them whenever you need them.  

If instead of trying to memorize the various properties, you take the 
time to make sure you can really follow these examples - and even 
explain them to someone else - then you should have no problems with 
exponents.  

I hope this helps.  Let me know if you'd like to talk about this some 
more, or if you have any other questions. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
Middle School Algebra
Middle School Exponents

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