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.
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!
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.
That wasn't too complicated, was it? But what if we saw this: 58/55. 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 58 and then 55. 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.
Let's cross out all five of the 5s on the bottom and five matching 5s on the top. That leaves us with one in the denominator and 5 * 5 * 5, or 53, in the numerator. Well, at least we got the one in the bottom bunk to take its hat off.
What's 5 * 5 * 5? Well, 5 * 5 is 25. And 25 * 5 is 125. So, 58/55 is 125. That was way simpler than figuring out 58 and 55, right?
It works with variables, too. Let's say we have this:
Ugh. It's like the Brady Bunch kids sharing a room, all with hats. Let's simplify. We can think of this as four separate fractions, one for each variable and one for the constants, 25 and 20 - those are like the Brady parents. At least they're not wearing hats.
Let's start with them, 25 and 20. Those simplify to 5 and 4. With the variables, we can just subtract the smaller exponent from the larger one. With a4 over a3, we're left with a over 1. No exponents! Awesome. Then there's b2 over b2. Well, those just go away, don't they? The middle child always seems to get left out. Okay, now c6 over c4. We simplify that to c2 over 1. If we put this all together, we have this: 5ac2/4. That's much, much simpler!
To summarize, we practiced solving exponential expressions. An exponent tells us how many times a number is multiplied by itself. We looked at positive and negative integers. With the latter, we were careful to include the negative sign when it was part of the expression.
We then looked at a problem involving a variable. Here, we plugged in the given value for the variable and solved. Finally, we looked at a couple of fractions. With these, we were able to first simplify the expression and then solve.
After watching this video lesson and reviewing its accompanying transcript, you could:
- Understand how to simplify and solve expressions with exponents
- Compare positive and negative integers
- Simplify expressions that contain fractions