Simplifying and Solving Exponential Expressions

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: Exponential Expressions & The Order of Operations

You're on a roll. Keep up the good work!

Take Quiz Watch Next Lesson
Your next lesson will play in 10 seconds
  • 0:03 Exponents
  • 1:00 Positive Integers
  • 2:16 Negative Integers
  • 3:38 Variables
  • 4:25 Fractions
  • 6:58 Lesson Summary
Add to Add to Add to

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Login or Sign up

Create an account to start this course today
Try it free for 5 days!
Create An Account

Recommended Lessons and Courses for You

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. 3^2? That's 3 * 3. 3^10? 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 7^3. 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, 7^3 = 343. We just solved it!

Let's try another expression with a positive integer: 3^10. 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. 3^10 = 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 x^4 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 6^4. That's 6 * 6 * 6 * 6. 6 * 6 = 36. 36 * 6 is 216, and 216 * 6 is 1,296. So, x^4 = 1,296 when x = 6.


That wasn't too complicated, was it? But what if we saw this: 5^8/5^5. Oh no. There are exponents in both parts of this fraction. It's like people in bunk beds wearing hats. I don't think you're supposed to wear a hat in bed. Let's simplify and solve so that no house rules are broken.

We could figure out 5^8 and then 5^5. Sure, that would work. And we'd get the correct answer. But simplifying first will save us some effort. Let's think about what we have here. The numerator is 5 * 5 * 5 * 5 * 5 * 5 * 5 * 5. The denominator is 5 * 5 * 5 * 5 * 5. That's a lot of 5s. But you know what? We can cross out the duplicates. 5/5 is just 1.

To unlock this lesson you must be a Study.com Member.
Create your account

Register for a free trial

Are you a student or a teacher?
I am a teacher

Unlock Your Education

See for yourself why 30 million people use Study.com

Become a Study.com member and start learning now.
Become a Member  Back

Earning College Credit

Did you know… We have over 95 college courses that prepare you to earn credit by exam that is accepted by over 2,000 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it free for 5 days!
Create An Account