Fractional Exponents

Uploaded by MATHRoberg on 22.11.2010

Hi, I'm Kendall Roberg, and today we're going to talk about exponents that are fractions.
Up here we have 25 to the 1/2 power.
What does that mean?
Well, it turns out 25 raides to the 1/2 power is the same as taking the square root of 25.
So what does the square root of 25 mean?
I've always thought of square roots as what number I would have to square to get the radicand.
So what number times itself...what number times itself...equals 25?
Well, 5 times 5 equals 25.
We call each of these fives a square root of 25.
So, the square root of 25 is 5, and also 25 to the 1/2 power is 5.
Let's try another one.
How about -27 to the 1/3 power?
Now, -27 to the 1/3 power is actually the same as finding the cube root of -27.
So now that we're dealing with a cube root instead of a square root, we have to think of what cubed would give us -27.
So what times what times what, or what times itself and times itself again, equals -27?
Well, with a little investigation and guess-and-check, you can probably come up with -3 times -3 times -31 equals -27, because -3 times -3 is 9 times another -3 is -27.
Let's try another one.
How about 32 to the 1/5 power?
Now what we have to take is actually the fifth root of 32.
And to answer this, we have to ask, "What raised to the fifth power would equal 32?"
So what times what times...well, what times itself times itself times itself again times itself one more time...equals 32?
And it turns out the number 2 times 2 times 2 times 2 times 2..well let's see, 2 times 2 is 4, 8, 16, 32.
So yes...32 to the 1/5 power is 2, and the fifth root of 32 is 2.
Oh, and I guess we never filled this one out.
-27 to the 1/3 power was -3.
The cube root of -27 is -3.
So there we go.
Now let's try one a little more complicated.
What if we're dealing with 4 to the 5/2?
Now, this is a little more challenging because the numerator of our fraction in our exponent is not 1, like it was in each of these situations.
If it was 1, it would be a lot easier to deal with.
So here's the trick: we can use the properties of exponents to change the numerator in our exponent to 1.
Let's rewrite 4 to the 5/2 as 4 to the 1/2 raised to the fifth power.
Now, if we simplify this, we'd end up back here, but when we have it written in this form, we can see that 4 to the 1/2 is the same as taking the square root of 4... this is the same as the square root of 4 raised to the fifth.
The square root of 4 is 2, so 2 raised to the fifth.
And we already found out that 2 raised to the fifth is 32.
So 4 to the 5/2 is 32.