EXPONENTS

What is Exponents?

An exponent is that little number written in superscript format at the top right position of a number or a variable. The exponent does nothing else that showing us how many times that number or variable is to be multiplied by itself. For example:

There are 5 theorems about exponents that will prove to be very useful in our work. Here they are:
 

If n and m are positive integer numbers and a is a real number, then: an X am = an+m If n and m are positive numbers and a is a real number: (an)m = anm If n es a positive integer and a and b are real numbers: (ab)n = anbn If n and m are positive integer numbers and a is a real number, there are three possible options: an/am  =  an-m        if n>m              =  1/am-n     if n<m              =  1               if n = m If n is a positive integer and a and b are real numbers, where b¹0, then: (a/b)n = an / bn

 

What happens if you have an exponent that is 0?
It's very easy. There is another theorem that explains that problem, and it is extended for negative exponents as well.
 

Theorems about Exponent 0 and Negative Exponents. a0 = 1 a-n = 1/an

Of course, none of the valid theorems in math or other sciences are true just because someone said so. Lets prove that any number powered by 0 equals 1.

Taking any number, for this example we'll take our friend X. What we have to prove is that X⁰=1. So, lets do it:
X⁰ = 1. Suppose that we have any exponent, lets take n.And if we take the fourth theorem about exponents we have: Xⁿ/X^m = X^n-m,

but we are to prove that X⁰=1 then, from that principle we can asume that m=n,
and our first sentence Xⁿ/X^m = X^n-m becomes Xⁿ/Xⁿ = X^n-n
but, the fourth theorem says that if n=m the operation Xⁿ/X^m = 1, and taking that the expression above becomes:
1 = X^n-n
But X^n-n = X⁰
And then we finished our proof
X⁰ = 1.