Evaluating quotient of fractional exponents | Mathematics I | High School Math | Khan Academy

Let's see if we can figure out what 256 to the 47th power divided by 2 to the 47th power is, and like always, pause the video and see if you can figure this out.

All right, let's work through this together. At first, you might find this kind of daunting, especially when you see something like 2 to the 47th power. Is that even, that's not going to be a whole number? How do I do this, especially without a calculator? I should have said do this without a calculator, but then the key is to see that we can use our exponent properties to simplify this a little bit so that we can do this on paper.

The main property that might jump out at you is if I have something, if I have x to the a power over y to the a power, this is the same thing as (x/y) to the a power. In our situation right over here, 256 would be x, 2 would be y, and then a is 47. So we can rewrite this; this is going to be equal to 256 over 2 to the 47th power.

This is nice; we're already able to simplify this because we know 256 divided by 2 is 128. So this is 128 to the 47th power. Now, this might also seem a little bit difficult. How do I raise 128 to a fractional power? But we just have to remind ourselves this is the same thing as 128 to the 17th power then raised to the 4th power. We could also view it the other way around. We could say that this is also 128 to the 4th power and then raise that to the 17th, but multiplying 128 four times, that's going to be very computationally intensive. Then we have to find the seventh root of that; that seems pretty difficult, so we don't want to go in that way.

But if we can get the smaller number first, what is 128 to the 17th power? Then that might be easier to raise to the fourth power. Now when you look at this and knowing that probably, uh, the question writer in this case—I'm the person who presented with you—is telling you that you're not going to use a calculator, it's a pretty good clue that, all right, this is probably going to be something that I can figure out on my own.

You might recognize 128 as a power of two, and maybe 2 to the 7th is 128. We can verify that. So let's see, 2 to the 1 is 2, 4, 8, 16, 32, 64, 128. 2 * 2 is 4, * 2 is 8, * 2 is 16, * 2 is 32, * 2 is 64, * 2 is 128. So 2 to the 7th power is equal to 128, or another way of saying this exact same thing is that 128 is equal to 2 to the 7th power.

Another way to say this is 128 to the 17th power is equal to 2, or you could even say that the 7th root of 128 is equal to 2. So we can simplify this; this is 2. So our whole expression is now just 2 to the 4th power. Well, that's just 2 * 2 * 2 * 2, so that's 2 to the 4th power, which is just going to be equal to 16. That's 2 * 2 * 2 * 2 right over there, and so we're done.

This crazy complicated-looking expression has simplified to 16.