Hello. I’m Professor Von Schmohawk and welcome to Why U.
In the previous lecture we saw that exponentiation is shorthand for repeated multiplication.
Any number "a" raised to the nth power
is equivalent to multiplying n a's together.
In an exponential expression,
"n" is called the "exponent" and "a" is called the "base".
We also saw that to multiply exponential expressions with the same base, we add their exponents,
and to divide exponential expressions with the same base, we subtract their exponents.
We can use these rules to find out what an exponent of one would mean.
If we start with a non-zero number "a", cubed
and divide it by the same number squared, we subtract the exponents,
and the result is "a" with an exponent of one.
We know that a number cubed
is equivalent to three copies of that number multiplied together,
and a number squared is equivalent to two copies multiplied.
We can simplify this fraction
by canceling out two copies of the number in the numerator and two copies in the denominator,
which leaves us with one copy of the number.
So any number raised to the first power is the same as the number itself.
To find out what an exponent of zero would mean,
let’s divide "a" to the first power by itself.
Subtracting the exponents, we get "a" with an exponent of zero.
Of course, any fraction where the numerator and denominator are the same is equal to one.
So we see that any number with a zero exponent is equal to one.
There is one caveat to be aware of.
What if "a" is 0?
We showed that "a" raised to the zero power is one
so you would think that zero raised to the zero power should be one.
The problem is that no matter how many times we multiply zero times itself,
we always get zero.
So is zero to the zero power one or zero?
The answer is that it is undefined.
So far we have seen the meaning of integer exponents of zero or greater.
But is there any meaning to a negative exponent?
Once again, we can use the rules of exponents
to determine the meaning of raising a number to a negative power.
If we take a non-zero number "a" raised to the zero power
and divide it by the same number raised to any positive power "n"
we subtract the exponents,
so the result will have an exponent of negative "n".
Since "a" to the zero power is equal to one,
this tells us that any number with a negative exponent
is equivalent to one over that number with the same positive exponent.
For example, "a" to the negative third power is equal to one over "a" cubed,
or one over "a" times "a" times "a".
The only number that cannot have a negative exponent is zero
since this would be the same as one over zero, and dividing by zero is meaningless.
So now we know the meaning of any integer exponent
and the rules of adding and subtracting exponents work in every case.
In our next lecture, we will see how exponents play a big role in our decimal number system,
and how they are used to create what is called "scientific notation".