Can a Number to the Zero Power be -1?

Date: 01/25/2002 at 17:06:44
From: Cheryl
Subject: A number to the zero power can be -1?

Dear Dr. Math,

My teacher told me that (-3^0) is equal to -1. He said that although 
anything to the zero power is one, in this case it is not so because 
the zero is INSIDE the parentheses with a negative number. Is this 
true?  

Thank you so much for your help.

Date: 01/25/2002 at 20:03:45
From: Doctor White
Subject: Re: A number to the zero power can be -1?

Hi Cheryl,

Your teacher is correct. Let me try to explain.

First, there is a difference between (-3)^0 and -3^0. 

Lets look at a different example. 

  4^3       -4^3        (-4)^3
 4x4x4     - 4x4x4      (-4)x(-4)x(-4)
  64        - 64           -64

In the second example the number is raised to the power, not the sign.
It could have been written as: -(4^3). This is read as the opposite of 
4 raised to the third power.

In the third example the parenthesis is raised to the power. This is 
read as a negative 4 raised to the third power.

Now back to our original problem: 

(-3)^0 is 1, but -3^0 can be written as -(3^0) which is the opposite 
of (3^0) which is -1.

I hope this helps you to see your teacher's solution. By the way, 
there is no difference between (-3^0) and -3^0. They both equal -1. 
The different one is (-3)^0, which equals 1.

Let me know if you need further assistance with this problem.

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

Date: 01/25/2002 at 21:44:39
From: Cheryl
Subject: A number to the zero power can be -1?

Dear Dr. White,

Thank you so much for your help! Your response was so fast and 
extremely clear. I am very grateful.

Sincerely,
Cheryl