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Working With Fractional Powers

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  • 0:03 Exponents
  • 0:52 Fractional Exponents
  • 3:09 Multiplying Fractional…
  • 4:23 Examples
  • 5:35 Lesson Summary
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Lesson Transcript
Instructor: Jeff Calareso

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

If you know what to do with 3 to the second power, then you understand exponents. But what if the exponent is a fraction? Find out what to do with fractional powers in this lesson.

Exponents

An exponent, also known as a power, looks like this: 5^2. The exponent tells us how many times a number or variable is multiplied by itself. So, 5^2 is 5 times itself, or 5 * 5. 5^3 is 5 * 5 * 5.

Think about bunnies. Who doesn't like thinking about bunnies? They're adorable. Well, they certainly must think so, because they have a bit of a reputation. You start with two bunnies, then you have four bunnies. That's 2 * 2, or 2^2. Then you have 8 bunnies. That's 2^3, or 2 * 2 * 2. Then you have 16 bunnies, which is 2^4, or 2 * 2 * 2 * 2. Then you decide bunnies aren't so cute anymore.

Fractional Exponents

But what if you have something like this: 9^(1/2)? How does that relate to bunnies?

This is a fractional exponent, or a fractional power. How do you multiply something by itself a fractional amount of times?

Think about what a fraction is. 1/2 is half of one. It's less than one. And a fractional exponent involves finding less than one of something. How much less than one? Well, you've probably seen 9^(1/2) before, though it didn't look like that. It looked like the square root of 9.

What is the square root of 9? 3. How do we know that? Because 3 * 3 = 9. And that's what 9^(1/2) is. It's the square root of 9.

When we have a fractional exponent, the denominator represents the root level. We can say that x^(1/n) is the nth root of x. So, 9^(1/3) is the cube root of 9. 9^(1/4) is the fourth root of 9.

Remember when we had 16 bunnies? If we do 16^(1/4), we find the fourth root of 16, which is 2. That's what happens when we give away all those bunny offspring to our unsuspecting friends. Like us, they'll need to learn about regular, whole number exponents before they learn about how awesome fractional exponents can be.

By the way, you may see a fraction where the numerator isn't 1. Don't worry; it works just like you'd expect. Look at x^(m/n). What is m/n? It's the same as m * 1/n, so that m works like a regular exponent. So, x^(m/n) = the nth root of x^m.

If we had 27^(2/3), we can think of it as the cube root of 27^2. 27^2 is 729. The cube root of 729 is 9. We could also think of it as square of the cube root of 27. The cube root of 27 is 3. 3^2? Yep, still 9.

Multiplying Fractional Exponents

If, for some reason, we decide we want to build a bunny farm, we're going to need to think big. That means multiplication and exponents. What happens when we multiply together two numbers with exponents, like 3^2 * 3^4? Do we get 3^8? No. One of the laws of exponents tells us that we add the exponents. So, 3^2 * 3^4 = 3^(2 + 4), which is 3^6. That's still a lot of floppy-eared cuteness.

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