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CCSS Math: HSN.RN.A.1, HSN.RN.A

We already know a good
bit about exponents. For example, we know
if we took the number 4 and raised it to
the third power, this is equivalent
to taking three fours and multiplying them. Or you can also view it
as starting with a 1, and then multiplying the 1
by 4, or multiplying that by 4, three times. But either way, this is
going to result in 4 times 4 is 16, times 4 is 64. We also know a little bit
about negative exponents. So for example, if I were take
4 to the negative 3 power, we know this negative
tells us to take the reciprocal 1/4 to the third. And we already know
4 to the third is 64, so this is going to be 1/64. Now let's think about
fractional exponents. So we're going to think about
what is 4 to the 1/2 power. And I encourage you
to pause the video and at least take a guess
about what you think this is. So, the mathematical
convention here, the mathematical definition
that most people use, or in fact that all people use here,
is that 4 to the 1/2 power is the exact same thing
as the square root of 4. And we'll talk in the
future about why this is, and the reason why this
is defined this way, is it has all sorts of
neat and elegant properties when you start manipulating
the actual exponents. But what is the
square root of 4, especially the
principal root, mean? Well that means,
well, what is a number that if I were to
multiply it by itself, or if I were to have
two of those numbers and I were to multiply
them, times each other, that same number,
I'm going to get 4? Well, what times
itself is equal to 4? Well that's of
course equal to 2. And just to get a sense of why
this starts to work out, well remember, we could
have also written that 4 is equal to 2 squared. So you're starting to see
something interesting. 4 to the 1/2 is equal to
2, 2 squared is equal to 4. So let's get a couple
more examples of this, just so you make sure
you get what's going on. And I encourage you to pause
it as much as necessary and try to figure
it out yourself. So based on what
I just told you, what do you think 9 to the
1/2 power is going to be? Well, that's just
the square root of 9. The principal root of 9, that's
just going to be equal to 3. And likewise, we
could've also said that 3 squared is, or
let me write it this way, that 9 is equal to 3 squared. These are both true statements. Let's do one more like this. What is 25 to the
1/2 going to be? Well, this is just
going to be 5. 5 times 5 is 25. Or you could say, 25
is equal to 5 squared. Now, let's think about what
happens when you take something to the 1/3 power. So let's imagine taking
8 to the 1/3 power. So the definition here
is that taking something to the 1/3 power
is the same thing as taking the cube
root of that number. And the cube root is just
saying, well what number, if I had three of that
number, and I multiply them, that I'm going to get 8. So something, times something,
times something, is 8. Well, we already know that 8 is
equal to 2 to the third power. So the cube root of
8, or 8 to the 1/3, is just going to be equal to 2. This says hey,
give me the number that if I say that number, times
that number, times that number, I'm going to get 8. Well, that number is 2 because
2 to the third power is 8. Do a few more examples of that. What is 64 to the 1/3 power? Well, we already know that
4 times 4 times 4 is 64. So this is going to be 4. And we already wrote over here
that 64 is the same thing as 4 to the third. I think you're starting to see
a little bit of a pattern here, a little bit of symmetry here. And we can extend this idea to
arbitrary rational exponents. So what happens if I were
to raise-- let's say I had, let me think of a good number
here-- so let's say I have 32. I have the number 32, and I
raise it to the 1/5 power. So this says hey,
give me the number that if I were to
multiply that number, or I were to repeatedly
multiply that number five times, what is that, I would get 32. Well, 32 is the same thing as
2 times 2 times 2 times 2 times 2. So 2 is that number, that
if I were to multiply it five times, then
I'm going to get 32. So this right over here
is 2, or another way of saying this kind of same
statement about the world is that 32 is equal to
2 to the fifth power.