Welcome to level three exponents.
Let's get started.
So if I asked you what 4 to the 1/2 power is, your immediate
inclination is to view this probably as like a
multiplication problem and try to multiply it somehow or add
them or something.
And you always have to remind yourself this
is not multiplication.
I know when I first learned it, I always was tempted to do
something with multiplication.
Well, something to the 1/2 power might not be intuitive to
you, but it actually turns out that this means the same thing
as the square root of 4.
Or another way, what times itself is equal to 4,
and we know that the square root of 4 is 2.
It could actually be a positive 2 or a negative 2 because we
know that either of those numbers when they're
squared could equal 4.
But for the sake of this one we'll assume it's always
the positive square root.
So 4 to the 1/2 is equal to 2.
Similarly, 9 to the 1/2, well that would be 3.
16 to the 1/2 -- oops.
16 to the 1/2, my subconscious gave away the answer.
16 to the 1/2 power is 4.
25 to the 1/2 power is 5.
I think that might make sense to you now.
So what does it mean when something is to the 1/3 power?
Well, if I say 8 to the 1/3 power, you might
already catch on.
In the 1/2 power we said something times something
is equal to 4.
Well in the 1/3 power we have to say that something to the
third power is equal to 8.
And if you've been practicing your exponents you know that 2
to the third power is equal to 8.
So we know that 8 to the 1/3 is equal to 2.
Similarly, 27 to the 1/3 is equal to 3.
And 64 to the 1/3 is equal to 4.
You might notice that I'm picking particular numbers,
like 8 and 27, 64.
That's because they have clean, cube roots.
And then by the way, when something is to the 1/3 power
that's the same thing as saying the cube root.
I just used terminology without explaining it,
which is very bad.
So I just used 8 and 27 and 64 because when I raised them to
the fractional exponents, they actually come out to
be clean numbers.
You could use a calculator and do something like 5 to the 1/3
power and you'll get some weird decimal.
Let's do some more problems.
So we know that 9 to the 1/2 is equal to 3.
Well what do you think 9 to the 2/3 is equal to?
Well, it turns out that this is equivalent to 9 to the -- oops,
I actually didn't want to do those -- what do you think 9
to the 3/2 is equivalent to?
Well this is the same thing as 9 to the 1/2 power
to the third power.
And I'll do a whole presentation on the actual
principles of exponents, but it actually turns out you
could just multiply.
When you have one exponent to another is when you can
multiply the two and that's where you get 3/2 .
But 9 to the 1/2 we know is 3.
And you're raising it to the third power, so that equals 27.
I'm sure at this point I have confused you.
Let's do more of these.
So you know at this point that 16 to the 1/4 power --
think about what that is.
That means that some number to the fourth power is 16.
If you've been practicing your level one exponents, you'll
probably know that well, that equals 2, because 2 times 2
times 2 times 2, well that equals 16.
So we know that 16 to the 1/4 is equal to 2.
So what do you think 16 to the 2/4 is equal to?
Well, we already know from that last problem that that's the
same thing as 16 to the 1/4 squared -- that's the 2 on both
sides -- and we know 16 to the 1/4 is 2, so that equals 2
squared and that equals 4.
And it all works out because we know from fractions another way
to write the fraction 2/4 is to write 1/2.
So this is the same thing as 16 to the 1/2 power.
16 to the 1/2 power, well that's just equal to 4.
Now I'm going to mix it up real good and do some negative
fractional exponents.
So what if I were to tell you 16 to the negative 1/2 power?
Well this might seem very daunting at first, but as we
know with the negative exponents level three,
immediately we just say well this is the same thing as 1
over 16 to the positive 1/2.
And that's the same thing as 1 to the 1/2 over 16 to the 1/2.
Well the square root of 1 is easy, it's 1.
And 16 to the 1/2 is 4.
So that wasn't too bad.
It's a little daunting when you see a negative exponent, but
immediately when you see that negative, just flip the 16 and
then work it out like a regular fractional exponent problem.
Let's do another one.
8 over 27 to the negative 1/3.
Immediately when we see that negative, we
want to just flip it.
So we'll say that equals 27 over 8 to the 1/3, and that
equals 27 to the 1/3 over 8 to the 1/3.
And we know that 27 to the 1/3, well that equals 3.
And 8 to the 1/3, well that equals 2.
So we've got 8 over 27 to the negative 1/3 is 3/2.
The first problem probably looked very intimidating to
you, but it only took us two steps to get there and as you
do more practice, hopefully it'll seem more and
more intuitive to you.
Let me give you another problem.
What's negative 8 to the negative third power?
Let me change that.
What's negative 8 to the negative 1/3 power?
Once again, at first this might confuse you, but when you see
that negative in the exponent, we just take the reciprocal of
the base, so that we say that that is equal to negative 1
over 8 to the 1/3 power.
And we say well that is equal to, we could write it as 1
over negative 8 to the 1/3.
We say what number times itself 3 times is equal to negative 8?
Well, we know from intuition there's no real mechanical way
to do this, but we know that negative 2 times negative 2 is
4 times negative 2 is negative 8.
So we know that this is equal to 1 over negative
2 or negative 1/2.
So negative eight to the negative 1/3 is
equal to minus 1/2.
Let's do another one, one more problem just to thoroughly
melt your brain.
Let's say 9 over 4 to the negative 3 over 2.
Well, immediately we see that negative exponent,
let's flip the base.
We get 4 over 9 to the 3/2 .
Well we know that that equals 4 over 9 to the 1/2, and all
of that to the third power.
4 over 9 to the 1/2, I think at this point you know that's the
same thing as 4 to the 1/2 which is 2 over 9 to
the 1/2 which is 3.
Now we have to just raise everything to the third power.
And that's the same thing as 2 the third power which is
8 divided by 3 to the third power.
Well that's 27.
There we have 9/4 to the negative 3/2 power is
equal to 8 over 27.
Now hopefully you can at least do these problems.
You probably don't have a good intuitive sense for exactly
what a negative 3/2 power is, and hopefully I can cover that
for you in future modules.
But I think you're ready to try some of the level three
exponent practice problems.
Have fun.