Simplifying and Solving Exponential Expressions

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  • 0:03 Exponents
  • 1:00 Positive Integers
  • 2:16 Negative Integers
  • 3:38 Variables
  • 4:25 Fractions
  • 6:58 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.

What do we do with an exponent? In this lesson, we'll learn how to simplify and solve expressions containing exponents. We'll solve a variety of types of exponential expressions.


What if you lived in a world where no one ever wore hats? This is what it's like for numbers in a world without exponents. Sure, sometimes their heads get wet. Plus, they don't have a discreet way to cover the fact that they didn't comb their hair.

Fortunately, some numbers have hats, which are called exponents. Now, exponents aren't just normal hats. They're worn off to the side, like a beret, to indicate the number's clear superiority over its hatless peers. But don't be intimidated by numbers with exponents. I mean, if you knew every person with a hat was just covering up messy hair, you wouldn't be intimidated, right?

An exponent is a number indicating how many times a number is multiplied by itself. 32? That's 3 * 3. 310? That's 3 * 3 * 3 *... well, times seven more 3s.

Positive Integers

So let's say you're asked to solve an exponential expression. Let's start with some involving positive integers, for example: 7^3. How do we solve this? Remember, the exponent tells us how many times we multiply the number by itself. You could think of it as: the bigger the number, the bigger the hat. And the bigger the hat, the more self-centered the number is, or the more time the number spends looking at itself in the mirror.

Our first expression is 73. That's not too fancy of a hat. We want to multiply 7 by itself three times. That's 7 * 7 * 7. Well, 7 * 7 = 49, and 49 * 7 = 343. So, 73 = 343. We just solved it!

Let's try another expression with a positive integer: 310. We saw this one before. That's a huge hat! This 3 must think it's pretty awesome. Let's count it out: 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3. Was that 10? Yep. If we do the math, we get 59,049. 310 = 59,049. That's huge!

Negative Integers

You don't have to have a positive outlook to wear a hat. Some of the gloomiest people I've ever met wore hats. The same is true with integers. Let's look at a couple of expressions with negative integers.

Let's start with (-4)2. That's a pretty modest hat. How do we solve it? Is it 4 * 4? No. Notice that the negative sign is inside the parentheses with the 4. It's in a glass case of emotion. So we need to do -4 * -4. That's positive 16. So, (-4)2 is positive 16.

What about (-1)3? Again, this expression is in a glass case of emotion. We want to do -1 * -1 * -1. We know it's going to be 1, but is it positive or negative? Let's see. A negative times a negative is a positive. And a positive times a negative is a negative. So, (-1)3 = -1.

With negative numbers, always check whether the exponent is odd or even. With odd exponents, the number will stay negative. With even exponents, the number will be positive. I guess odd hats can keep you feeling negative while even hats even you out.


Let's move on from these hat-wearing negative integers and try some variables. You may encounter a problem like this: Solve x4 when x = 6. What's this? This is like seeing a hat on a mannequin. x is our variable, blankly carrying the exponent while we figure out if we'd look good wearing it. Hey, mannequins look at themselves in mirrors, too. Well, they would if they had eyes.

Fortunately, we're told that x = 6. So what we need to figure out is 64. That's 6 * 6 * 6 * 6. 6 * 6 = 36. 36 * 6 is 216, and 216 * 6 is 1,296. So, x4 = 1,296 when x = 6.

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