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Algebra I: High School20 chapters | 168 lessons | 1 flashcard set

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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.

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.

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 *n*th 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 *n*th 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.

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.

But we don't just raise bunnies to marvel at their multiplying powers. We also sell them as pets. Just watch out for the fidgety one with the stopwatch. But reducing our bunny population means fractional exponents.

When we **multiply fractional exponents**, we still add the exponents. So, 3^(1/2) * 3^(1/2) = 3^(1/2 + 1/2). What's 1/2 + 1/2? 1. So, 3^(1/2) * 3^(1/2) = 3^1, which is just 3. Think about that for a second. We're saying the square root of 3 times the square root of 3 is 3, which makes sense, right?

Let's look at a few more examples. What about 144^(1/4) * 144^(1/4)? We want to add those 1/4ths together to get 2/4, which is 1/2. So, we have 144^(1/2). That's the square root of 144, which is 12. Have you ever counted 144 bunnies? That would make a lot of rabbit stew...Oh, I'm sorry.

What about 64^(1/9) * 64^(2/9)? Well, 1/9 + 2/9 is 3/9, or 1/3. So, we want the cube root of 64, and that's 4.

Let's try one more: 64^(1/3) * 64^(1/3). Well, 1/3 + 1/3 = 2/3. So, we have 64^(2/3). Remember, we can think of this as the cube root of 64^2 or the square of the cube root of 64. Well, the cube root of 64 is 4. And 4^2? That's 16. So, 64^(2/3) = 16.

To summarize, we started with a refresher on exponents. An exponent tells us how many times a number is multiplied by itself. Think of bunnies multiplying exponentially.

We then learned about fractional exponents. *x*^(1/*n*) tells us to find the *n*th root of *x*. So, 81^(1/3) would be the cube root of 81.

The numerator represents our whole number exponent, so *x*^(*m*/*n*) is the same as the *n*th root of *x*^*m*.

When we multiply exponents together, the law of exponents instructs us to add the exponents. 5^3 * 5^3 is 5^6.

With fractional exponents, the same law applies. 16^(1/4) * 16^(1/4) is 16^(1/4 + 1/4), which is 16 to the 2/4 power, or 1/2.

We also may have learned that bunnies aren't just cute, they're a floppy-eared industry.

After you've reviewed this video lesson, you will be able to:

- Explain what to do with numbers when there is a fractional exponent
- Multiply numbers with fractional exponents

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Algebra I: High School20 chapters | 168 lessons | 1 flashcard set

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