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0^0 is Undefined


What does 0^0 equal?

Some people think 0^0 equals 1, while other people think 0^0 is undefined. Here is a simple proof to answer this mystery.

Assume that:


x^0 = y


Take the natural log or ln of both sides of the equation:

ln (x^0) = ln y


Since ln (a^b) = b * ln a, then this is true:

0 * (ln x) = ln y


When x is greater than or less than zero (x != 0), then (ln x) is a number (n), so this is true:

0 * n = ln y         when x != 0


Thus:

0 = ln y         when x != 0


If we apply Euler's number (e) to both sides of the equation, this would be the result:

e^0 = e (ln y)         when x != 0


Simplified, this is:

1 = e (ln y)         when x != 0


More simply:

1 = y         when x != 0


Therefore, if we substitute this equation into the original assumption., then we would have proved:

x ^ 0 = 1         when x != 0


Thus, any non-zero number to the zeroth power equals 1.




However, when x is equal to zero (x=0), then (ln x) is (ln 0), and this equation is undefined.

Since we had this equation above:

0 * (ln x) = ln y


We can substitute to this equation when x is equal to 0.

0 * (ln 0) = ln y         when x = 0


This equals:

0 * undefined = ln y         when x = 0


Therefore, we must stop working on this equation. As a result:

x^0 = undefined         when x = 0


or

0^0 = undefined


by Phil for Humanity
on 09/14/2011

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