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
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.
Exponent 0 and Negative Exponents.
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/Xm = Xn-m,
but we are to prove that X0=1 then,
from that principle we can asume that m=n,
and our first sentence Xn/Xm = Xn-m becomes Xn/Xn = Xn-n
but, the fourth theorem says that if n=m the operation Xn/Xm = 1, and taking that the expression above becomes:
1 = Xn-n
But Xn-n = X0
And then we finished our proof
X0 = 1.