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Up until this point we've only dealt with exponents that are either
positive or negative integers.
Today we're going to look at exponents of another form.
I'd like to look at exponents in the form of 1/n.
Say we have x raised to the 1/n, we can rewrite that as
the nth root of x.
Where x to the 1/n is written in exponential form
and the nth root of x is written in radical form.
These two expressions are equal.
And I also wanted to point out that this n
is the index of the radical.
So let's explore. We have 8 raised to the 1/3.
And you can see that my x is 8 so
where I have x, I replace that with an 8.
My n is 3 so I replace the n with 3.
And now I have the cube root of 8.
Now we know the cube root of 8 is 2
because this is really asking me what number multiplied by itself
three times, give me 8. The answer is 2.
Because 2*2*2 is indeed 8.
We can verify this with our graphing calculator
by entering 8 raised to the (1/3), in parenthesis.
Right, so 8 raised to the (1/3) is the same as
8 to the 1/3, we press enter,
and we have 2.
We can also enter this expression into our calculator.
If I press 3, that's for the cube root.
And I press the math key.
You see where we have number five?
I'm going to arrow down to highlight number five.
That's saying my index is x.
Well we have already entered 3 for my index.
So I press the five key.
And you see now, I'm taking the third root of...
I'm going to take the third root of 8.
So this, it looks a little different than what I have
written on the screen, but this is essentially the same expression.
The cube root of 8, or the third root of 8.
I press enter... ...and I get 2 as well.
Now let's take a look at exponents of the form m/n.
That's m as in Mary over n as in Nancy.
We can write this two ways.
Option A: x to the m/n is the
nth root of x all raised to the mth power.
Option B is the nth root of x to the m.
So it doesn't matter if I take this whole expression
and raise it to the mth power or
if I actually put that underneath the radical symbol.
We're going to do a few examples both ways
to verify that the answer should still be the same.
Before I move on, one thing I want to point out.
A lot of my students mix up the m and the n.
They're not sure if the m goes here or the n goes here.
Just think, if this was a flower, the root would be on the bottom.
And the flower, or the power, would be on the top.
So the root, n, goes on the bottom
and the power, m, goes on the top.
Again, this is written in exponential form
and this is written in radical form.
So your teacher might give you one or the other
and ask you to write in a different form.
So I might give you something in exponential form
and say, "rewrite in radical form."
Or I could give you something in radical form
and ask you to rewrite in exponential form.
You should feel comfortable doing this.
So using option A, if I were to give you 8 raised to the 2/3,
well this 2/3 can be broken apart, right?
1/3 times 2 is still 2/3.
So now we have 8 raised to the 1/3 and the power is on the outside,
so this is all raised to the second.
And think about it, 2*1/3 is 2/3.
So we really haven't changed anything here,
we've just rewritten it several times.
Now, remember, in parenthesis here,
8 to the 1/3 can be rewritten as, the cube root of 8,
and that's all squared.
So if you look at this as the 2 = m,
the m was on the outside, like you see here.
The n is 3, that does in the index, just like you see the n here.
And x stays where it is.
So the 8 stays underneath the radical symbol.
So when we actually evaluate this
the cube root of 8 is 2,
2 squared is 4.
Remember, this is asking you,
what number mulitplied by itself three times gives you 8?
That answer is 2. Then 2 squared = 4.
We can verify this result using our calculator.
So first let's enter the original problem.
We had 8 raised to the (2/3).
Press enter.
My calculator gives me 4.
Now to enter this in radical form,
well, we're going to have parenthesis because
this first piece is going to go in parenthesis.
The index is 3, I press the math key,
number five.
So now I have the third root of 8.
I'm going to close my parenthesis, and this is all squared.
So it's all raised to the second.
Everything inside the parenthesis is raised to the second.
I press enter and we get 4.
Now let's look at option B.
This time I'm putting the 2 in front of the 1/3.
I'm doing 2*(1/3).
In the previous example I did (1/3)*2.
So when I break this apart to 2*(1/3), this becomes
8 raised to the second, in parenthesis,
and the 1/3 is on the outside.
Again, if I multiply 2*(1/3) I get 2/3.
So now I'm squaring the 8
and taking the cube root of everything inside the parenthesis.
So when we clean this up
8 squared is 64.
What number multiplied by itself three times would give me 64?
Well 4*4 is 16, 16*4 is 64.
Again to verify our answer, we pull up our trusty calculator.
And I'm going to enter exactly what I see here.
So the third root of, let'*** the math key.
Number five.
Now it's 8 squared, so I'm going to put this in parenthesis.
8 raised to the second power.
So this looks like what I have here.
The third root of 8 squared. The third root of 8 squared.
I press enter, and we get 4.
So we're in business.
If we look options A and options B side by side
here you can see the, everything in parenthesis is squared.
And in option B I actually brought the second power underneath
the radical symbol.
And either way we do this, we get the same answer.
The big thing is to make sure you have the correct index.
If you feel a little fuzzy about this,
the next video I'm actually going to do a few problems
on the blackboard, by hand.
So look for the videos that have a "BB" next to them.
That indicates "black board" or black board example.
Thank you for watching, and take care.