How Can a Number Raised to the Zero Power Be One?

Date: 01/23/2008 at 14:22:19
From: Aron
Subject: WHY is any number raised to a power of  zero, one

I understand that 3 raised to the 7th multiplied by 3 raised to the 0
is equal to 3 raised to the 7th, because the rule is that you would
add the exponents which would give you 3 to the 7th which clearly
alludes to the fact that 3 to the zero equals one so that 3 to the 7th
is equal to itself........but WHY?  

How, mathematically is 3 to the zero equal to one????

Date: 01/23/2008 at 15:02:24
From: Doctor Peterson
Subject: Re: WHY is any number raised to a power of  zero, one

Hi, Aron.

Our FAQ page has several explanations:

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

Here is another way to make sense of it:

Rather than think of x^n (that is, x raised to the nth power) as

  x^n = x * x * ... * x
        \_____________/
         n copies of x

we can think of it as starting with 1 and then doing n 
multiplications--literally "multiplying by x, n times"

  x^n = 1 * x * x * ... * x
          \_______________/
        n multiplications by x

Taking it this way, we have:

  x^2 = 1 * x * x
          \_____/
        2 multiplications

  x^1 = 1 * x
          \_/
        1 multiplication

  x^0 = 1
          \/
        0 multiplications

So x^0 is 1, for any x.

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


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 01/23/2008 at 15:36:44
From: Aron
Subject: Thank you (WHY is any number raised to a power of  zero, one)

Thank you - your answer is awesome!  Aron