# Powers of 0 and 1

• Jul 18th 2008, 02:55 PM
magentarita
Powers of 0 and 1
We all know that any number or variable raised to the zero power equals 1
and that any number or variable raised to the 1st power equals the number or variable.

SAMPLE A:

x^0 = 1 as well as 100^0 = 1

SAMPLE B:

x^1 = x as well as 100^1 = 100

Why is this the case?

Can anyone explain, in simple terms, the WHY concerning the above?
• Jul 18th 2008, 03:05 PM
arbolis
I'll do the why of
Quote:

x^0 = 1
. We have $x^0=x^{1-1}=x^1\cdot x^{-1}=\frac{x^1}{x^1}=1$. EDIT : Or even better, $x^0=e^{0\cdot\ln x}=e^0=exp(0)=1$ because $\ln(1)=0$ (From the definition of the logarithm function. If you define the exponential function as the inverse function of the logarithm one, then what I wrote holds.). Note that it holds $\forall x \neq 0$.
• Jul 18th 2008, 03:12 PM
Mathstud28
Quote:

Originally Posted by magentarita
We all know that any number or variable raised to the zero power equals 1
and that any number or variable raised to the 1st power equals the number or variable.

SAMPLE A:

x^0 = 1 as well as 100^0 = 1

SAMPLE B:

x^1 = x as well as 100^1 = 100

Why is this the case?

Can anyone explain, in simple terms, the WHY concerning the above?

Assuming you buy Arbolis's post then

$x^1=\frac{x^{a+1}}{x^a}$

Or more obvious

$\frac{x^2}{x}$

Assuming $x\ne{0}$
• Jul 19th 2008, 06:00 AM
magentarita
Thanks A Million!
I thank you both for that great mathematical insight.