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More on Why n^0 = 1

Date: 11/03/2007 at 20:46:20
From: Connor
Subject: why does n^0=1 (different from other question you got)

Why does n^0 = 1?  At first my question may seem the same as the other
question in your archive, but read the whole thing please, because
it's different.  I saw your other question on n^0, and you gave the
same response as the one I got from my math teacher.  My question,
which she couldn't answer, is if 2^3=2*2*2, 2^2=2*2, and 2^1=2,
shouldn't, 2^0=  , meaning that 2^0=0?

I already know that 2^5=2^6/2 and so on.  I've tried 2^3=1*2*2*2, but 
it didn't seem to make sense to have to add that 1. (In that case 
2^2=1*2*2, 2^1=1*2, and 2^0=1.)



Date: 11/03/2007 at 22:11:52
From: Doctor Peterson
Subject: Re: why does n^0=1 (different from other question you got)

Hi, Connor.

We get LOTS of questions about this, and there is more than one answer
to be found in our archive.  I'll suppose that what you found is the FAQ:

  N to 0 power
    http://mathforum.org/dr.math/faq/faq.number.to.0power.html 

My way of explaining that fits the form of your question nicely.  It
starts with the observation that talking about a power a^n as 
"multiplying a number by itself n times" is very awkward, since really
there are n-1 multiplications!  You have to say something like
"multiply together n copies of the base".  But there's a way to say it
that does involve n multiplications: start with 1, and multiply BY the
base n times:

  a^n = 1 * a * a * ... * a
          \_______________/
               n times

This is the definition you mention; I don't see why you say it doesn't
make sense.  It clearly gives the same result for positive integer
exponents, since multiplying anything by 1 doesn't change it; and it
provides a neater way to state the definition.  Does what I said to
introduce it help it make more sense to you?

Now, when you see it this way the answer becomes obvious:

  2^3 = 1*2*2*2 = 8
  2^2 = 1*2*2   = 4
  2^1 = 1*2     = 2
  2^0 = 1       = 1

The answer falls right out of the definition this way.

Note that when you are doing addition, a blank ("adding nothing")
means zero.  But when you are doing multiplication, a blank
("multiplying nothing") means 1!  The same thing happens when you
cancel in a fraction:

  / /
  2*3      ?      
  ----- = ---
  2*3*5    5
  / /

What's left when you've canceled everything?  Not 0, but 1, since it's
really

    / /
  1*2*3      1     1
  ------- = --- = ---
  1*2*3*5   1*5    5
    / /

So the role of 1 here is common throughout algebra.

Now, do you know about negative exponents?  How could you extend this
definition to cover that case?

If you have any further questions, feel free to write back.

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

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