Hello.
I’m Professor Von Schmohawk and welcome to Why U.
In the previous lecture, we saw that any number with an exponent one over n
represents the nth root of that number.
In this lecture, we will show that
exponents can also be fractions with numerators other than one.
In fact, an exponent can be any positive or negative rational number.
To illustrate how this works, let’s start with the cube root of "a"
which can be written as "a" to the one-third power
and then square it.
As we saw in the chapter on Raising Exponential Expressions to Powers
we can simplify the expression on the right
by multiplying the exponents one-third and two
which gives us "a" to the two-thirds.
So "a" to the two-thirds represents the cube root of "a" squared.
However, we could have also come up with "a" to the two-thirds in a different way.
Instead of squaring the cube root of "a"
we could have first squared "a" and then taken the cube root.
Using exponents, this would be written as "a" squared to the one-third power.
And multiplying the exponents, once again we get "a" to the two-thirds power.
So in both cases we get "a" to the two-thirds power
regardless of whether we start by taking the cube root of "a", and then squaring it
or start by first squaring "a" and then taking the cube root.
It may seem strange that both these operations give the same result.
However, if we pick a value for "a" which comes out even
it is easier to see why this is true.
For example, let’s say that "a" is equal to eight.
Now on the left side
the cube root of eight is two, and two squared is four.
While on the right side
eight squared is 64, and the cube root of 64 is also four.
If this still seems a bit like magic, let’s look at it another way.
Let’s start by writing the number eight as two times two times two.
When we take the cube root, this leaves us with one two.
Then squaring this, gives us two twos.
On the right side
we again start by writing eight as two times two times two.
We then square this quantity, giving us six twos.
Then taking the cube root, leaves us with two twos.
So in both cases, we end up with two twos multiplied together, which equals four.
It is now easier to see why both operations produce the same result.
On the left side, we first eliminate all but one-third of the factors
and then multiply the number of factors by two
while on the right side we first multiply the number of factors by two
and then take one-third of the factors.
Since it doesn’t matter in which order we multiply the number of factors
by one-third and two
either way we will end up with the same number of factors, and will get the same result.
So we have two options for evaluating an expression with an fractional exponent.
We can raise the base to the power specified by the numerator
and then take the root specified by the denominator.
Or the operations can be performed in the opposite order
taking the root first and then raising the result to a power.
Either way we get the same answer.
However, the amount of work that we have to do can be a lot different.
Let’s do this example both ways and see why.
If we square 27 first, we get 729.
Then, taking the cube root of 729 gives us nine.
On the other hand, if we take the cube root first, we get three.
And squaring three, gives us nine.
Either way we get nine.
However, the second way is a lot easier.
In the beginning of the lecture, we said that an exponent can be any rational number
and in general, we can write this as the integers m divided by n.
But what would a negative rational exponent mean?
If we take a non-zero number "a" raised to the zero power
and divide it by the same number with a rational exponent m over n
we subtract the exponents, so the result will have an exponent of negative m over n.
Since "a" to the zero power is equal to one
this tells us that any number with a negative rational exponent
is equivalent to one over that number with the same positive rational exponent.
Rational exponents follow all the same rules we already know for integer exponents.
For example, in an expression written as a fraction
we can move the exponential terms from the denominator to the numerator
or from the numerator to the denominator
as long as we switch the sign of its exponent.
As an example
let’s say that we want to write the expression one over the square root of x
without using a radical sign.
We can eliminate the radical sign
by writing the square root of x as x to the one-half power.
But then we can simplify this expression further
by moving the x term from the denominator to the numerator
and switching the sign of the exponent.
So one over the square root of x, can be written as x to the negative one-half power.
Radical expressions can often be simplified by using the rules of exponents.
For instance, let’s simplify the expression "a" over the square root of "a".
We start by writing the square root of "a" as "a" to the one-half.
If we write the "a" in the numerator as "a" with an exponent of one
we can subtract the exponents giving us "a" to the one-half
or the square root of "a".
So "a" over the square root of "a"
can be more simply written as the square root of "a".
As another example, let’s simplify the square root of "a" over the cube root of "a".
We start by writing the square root of "a" as "a" to the one-half
and the cube root of "a" as "a" to the one-third.
We can then subtract the exponents one-half and one-third.
Using a common denominator of six, we write this as three-sixths minus two-sixths
which gives us "a" to the one-sixth or the sixth root of "a".
So the square root of "a" over the cube root of "a"
can be more simply written as the sixth root of "a".
Let’s do one more example
simplifying the cube root, of "a" squared, over the sixth root of "a".
We can start by writing the cube root of "a" squared, as "a" to the two-thirds
and the sixth root of "a" as "a" to the one-sixth.
We then subtract the exponents two-thirds and one-sixth.
Using the common denominator six, we can write this as four-sixths minus one-sixth
which is three-sixths, or one-half .
Of course, "a" to the one-half is the square root of "a".
So the cube root, of "a" squared, over the sixth root of "a"
can be simply written as the square root of "a".
So, we have seen that roots can be written
as exponential expressions using rational exponents
or as radical expressions using a radical sign.
We have also seen that these expressions can be simply written as an integer
if they are perfect roots.
In the next lecture, we will see that these expressions can sometimes be simplified
even when they are not perfect roots.