# Why is anything to the power zero one?

Zero exponent Here's a fairly rational argument. If we have any number x, to raise its current exponent (or power) by 1 we multiply by x. xa+1 = x(xa) For instance, you'll agree that 33 = 32 x 3 ; 27 = 9 x 3. It follows that to lower the exponent of x by one we divide by x. xa-1 = (xa) / x

e.g. 33 = 34/3 ; 27 = 81/3 This holds when the original exponent, a, is 1. x0 = x1/x = 1

e.g. 30 = 31/3 = 1

Anything to the zero power equals one because in multiplication and division, the only way to get zero is to either multiply by zero or to divide zero by something. When you are working with powers, you are multiplying. You are not multiplying by zero when raising to the zero power. Multiplying to the zero power is a expression indicating that you divide that number by itself, rather than multiplying by itself. It's like this:

3^0=3/3=1

3^1=3=3

3^2=3x3=9

3^3=3x3x3=27

etc...

Consider the law of exponents:

nx/ny=n(x-y)

This works for all numbers and powers. Therefore

33/32=3(3-2)=31=3 or

33/32=27/9=3

So

33/33=3(3-3)=30=1 is true as

33/33=27/27=1