Expert: Paul Klarreich - 2/13/2006

Question
i  am  a  8th  std student. x to the power0=1[xo=1]. how  itpossible  when any number is multiplied by 0=0.

Answer
Hi, Kiruthika,

You wrote:

Question:  i am a 8th std student. x to the power 0 = 1[xo=1]. how is it possible when any number is multiplied by 0=0.
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Well, there are two issues here.

1. The Power definition:  x^0 = 1  (observe notation) that you wrote is not totally accurate.  It should say:

For all x /= 0,  x^0 = 1.

[Notation:  /=  means 'not equal to'.]

2. Why should that be the definition?  Observe, by the way, that this is a DEFINITION, not a conclusion.  But choosing  x^0 to be equal to 1 is done so that it is consistent with other exponent rules and with other algebraic expressions.  For example:

--------- FIRST EXAMPLE
A. What is the rule for dividing power of x?  For example,  x^12/x^4.  You have been taught to keep the base, x, and subtract the exponents.  So the answer is x^8.  Then what is  x^5/x^5?

1) If we use the exponent rule, the answer is x^0.
2) If we use what we were taught earlier (in the 7th std?), dividing something by itself gives an answer of 1.

Are we supposed to get different answers depending on how old we are?

---------SECOND EXAMPLE -------
B. What does  x^3 actually mean?  The PROPER way to interpret this is to say 'three factors of x'.  [Do not say things like 'multiplied by itself'.]    So how do we evaluate  7x^3 ?  We write the coefficient, followed by three factors of x:

7x^3 = 7 x x x

and then do whatever is necessary.

Then what does 7x^0 mean?  Again, if we want to be consistent, it means to write the coefficient, followed by zero factors of x.  So after we have written the 7, we are done:

7x^0 = 7

What number behaves like this?  You multiply it by 7 and the answer is 7.  You guessed it -- the number 1 behaves like this.  So since x^0 behaves like the number 1, it is proper to define it to be 1.
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So you can define x^0 to be anything you like, but if you don't define it to be equal to 1, all our other rules have to change; they have to have exceptions, and math would be a lot harder.