Changing Negative Exponents to Fractions

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  • 0:03 Exponents
  • 0:44 Negative Exponents
  • 2:30 Practice
  • 4:30 Variables
  • 5:56 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.

A negative exponent isn't like a normal exponent. In this lesson, we'll learn about negative exponents and how they lead us to fractions. We'll practice with numbers and variables.


We define an exponent as a^n. That means a is multiplied by itself n times. It's like you build a human copy machine. If you go in, two of you come out. If two of you go in, four of you come out. If four of you go in, 16 of you come out. Whoa, that's a lot of you.

What's happening with your machine? It's multiplying you by you. In pure numbers, 2^2 is 2 * 2, or 4. 3^4 is 3 * 3 * 3 * 3, which is 81. Imagine having 81 copies of you. You could be your own marching band!

Negative Exponents

But what about this: a^-n? -n? That's weird. It's also a negative exponent. If a positive exponent multiplies a number by itself, what do you think a negative exponent does? Does it multiply a number by itself while making it feel sad? No. It's the opposite of multiplication: division.

A negative exponent leads to the inverse of a number. In other words, a^-n = 1/a^n. 3^-2 isn't 3 * 3, but 1/3^2. This is how we change negative exponents to fractions.

3^2 1*3*3 9
3^1 1*3 3
3^0 1 1
3^-1 1/3 .333...
3^-2 1/3/3 .111...

A great way to remember this is to always start with 1. Why? Well, 3^0 is just 1. 3^1 is like 1 * 3, or 3, which is 3 times larger than 3^0. 3^2 is like 1 * 3 * 3. That's 9, or another 3 times larger.

That's going from 0 to 1 to 2. Now let's go from 0 to -1. 3^-1 is 1/3, or .333 repeating. That's 3 times smaller than 3^0. 3^-2 is 1/9, or .111 repeating. That's 3 times smaller than 3^-1.

Remember how we used our magical machine with positive exponents to make copies of ourselves? What if there's a dial on it that we switch the positive exponent to a negative? Oh, that wouldn't be good. That'd be like a teleporter going awry. If three of you go in, 1/9 of you comes out. You've become a fraction! I hope it's the best 1/9 of you.


Let's practice with a few negative exponents to get comfortable with them. What about 2^-4? So we take the inverse of 2 to the positive fourth power, or 1/2^4. 2^4 is 2 * 2 * 2 * 2, or 16. So 2^-4 is 1/16.

What about 8^-1? First, turn it into a fraction, which makes the exponent positive: 1/8^1. What's 8 to the first power? It's just 8. So our answer is 1/8

Oh, man, I hope these aren't all people using a copy machine on the wrong setting.

What about 6^-3? Let's make it a fraction. It's 1/6^3. 6 * 6 is 36, and 36 * 6 is 216. So 6^-3 is 1/216. Hmm, 1/216 of something is really not very much. If you put a cat in there, you'd maybe have a whisker. Sorry, Mittens!

But wait. What if you start with a fraction, like 1/4^-3? What's the inverse of an inverse? Well, it's 1/1/4^3. You can't have a fraction like that. This isn't 3D chess. 1/1/4^3 is 4^3/1, or just 4^3. And 4^3 is 64. So our answer is 64. So a negative exponent in the denominator of a fraction gets us back to big numbers. Instead of a cat's whisker, we're now in crazy cat lady territory.

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