Question

Why X to the power of zero equals 1? Could somebody explain to me please?

Answer 1

This explanation is from Aldo Daniel Completa, a contributor from Argentina:

Let's begin with examples.

Ex.1: 5^3 / 5^2 = 125 / 25 = 5
5^3 / 5^2 = 5 ^(3-2) = 5^1 = 5

Ex.2: 5^3 / 5^3 = 125 / 125 = 1
5^3 / 5^3 = 5 ^(3-3) = 5^0 =


... and the result must be 1. So 5^0 =1.
The rule is: x^b / x^c = x^(b-c).

In order to generalize this rule for the case b=c, it must be defined that x^0 = 1 (x is any number different from 0). In mathematics, usefulness and consistency are very important. This convention allows us to extend definitions of power that would otherwise require treating 0 as a special case.

This method can also explain the definition: x^(-b) = 1 / x^b.

Ex. 5^2 / 5^4 = 25 / 625 = 1 / 5^2
5^2 / 5^4 = 5 ^(2-4) = 5^(-2)

http://mathforum.org/dr.math/faq/faq.number.to.0power.html

Answer 2

It's very simple. It equals 1 because it is defined that way in order for consistency. There are two rules: A number divided by itself =1 and when you divide you subtract exponents. So in order for both of these to be true, a^0 must be defined as 1.

Answer 3

\( x^{-n}=\dfrac{1}{x^n}\)

\(x^a\cdot x^b=x^{a+b}\)

Now

\( \dfrac{2^3}{2^3}=\dfrac{2\cdot 2\cdot 2}{2 \cdot 2\cdot 2}=1\)

\(\dfrac{2^3}{2^3}=2^3\cdot 2^{-3}=2^{3-3}=2^0\)

What should \(2^0\) be?

Answer 4

1. Thanks, everyone.