Why does X to the power 0 equal 1?

Answer 1

Applying the laws of exponents, xy / xy = xy-y. Therefore, if x2 / x2 = 1, then x2-2 = 1.

Or how about this, if it is easier to understand:

Logically there should be no such thing as something to the nothing power anymore than we can envision anything that is nothing (try it). But for the purpose of mathematics, 1 creates a continuous graph between the positive powers and the negative powers with no void in between.

As follows:

2p1 = 2, 2p2 = 4, 2p3 =8 and so on, so you can see that the result, going up is simply 2 times the previous result and going down is simply the previous result divided by two. This makes 2p0 equal to 1 (by dividng the previous result by 2). This continues into the negative powers since 2p-1=1/2 (half the previous result) and 2p-2=1/4 (again half the previous result) and so on.

This is true for all numbers, therefore xp0=1.

It is not logical, it is convenient for mathematical calculations...and it works!!

Answer 2

x3 = x times x times x

x0 = x divided by x ---- (another explanation)

x0 = 1 provided x in not 0 because this definition is consistent with all other definition of exponents. The easiest is the rule for multiplying powers of a like base by adding the exponents:

(xp)(xq) = xp+q Suppose q = 0 and you apply the rule to get

(xp)(x0) = xp+0 = xp (1) . Cancel xp from both sides and get x0 = 1.

Another rule says to divide powers, subtract the exponents:

(xp)/(xq) = x p-q

Suppose you apply the rule when p = q: You get

(xp)/(xp) = x p-p = x0. But xp/xp = 1 so x0 must be 1.

A more complicated reason is that the limit as x goes to infinity of x(1/n) is 1.

so it makes sense to define x0 = 1.