Everyone, so a lot of times in our algebraic expressions, we'll see the same thing, the number or a variable that gets multiplied by itself over and over again. And this is super inefficient and annoying to have to write out. What I'm going to do in this video is I'm going to show you that we have a special notation for writing this called exponent notation. So I'm going to show you that this 4 times itself a bunch of times can actually just be written as 45. That's what I'm going to show you in this video, just exponents and expressions. Let's go ahead and get started here. So, basically, we use exponents to represent repeated multiplication. So, for example, I had 4 times itself 5 times, 4, 1, 2, 3, 4, 5. And so, basically, what this 4 represents is the 4 represents the base. It's the number or, in some cases, it could be a variable. So number or variable that's being multiplied a bunch of times, and it's raised or sorry, and it is multiplied 5 times, that's what we call the exponent or the power. It's basically the number of times that base is being multiplied. So we write this as 45, and we actually say it as it's 4 raised to the 5th power. Alright? So that's all an exponent is, is it just says this thing is multiplied by itself a bunch of times. Alright? Now in some cases, we want to have to condense, a bunch of, you know, numbers into a smaller format. We'll actually have to expand it out and see what all the multiplication is. So, for example, if I have something like x to the exponent of 3, that just means x×x×x. Right? So we can expand it and condense it as well. And, by the way, so the x the base here is just x, and the exponent is 3. And one of the other ways you might hear that is you might hear something like x cubed. That's what that third power means. So, basically, the general form of any exponent is if I have something a, a is just a generic letter. It could be a number or a variable that's multiplied by itself a bunch of times, so in other words, there's, like, a there's a bunch of a's here, then that just means I can say that this is an raised to the nth power. That's the general notation for this. Alright? So other than that, that's really all there is to it. So let's just go ahead and take a look at some problems now because now our algebraic expressions are going to involve some exponents, but, really, we're going to be doing the same thing. We're going to be evaluating expressions. We know exactly how to do that. So let's start with our first problem here. We have negative 3x to the 4th power. And if we want to evaluate this algebraic expression, remember, we just replace letters for numbers. Every time I see an x, I replace it with a 2. So, for example, in this problem here, I have negative 3, and then I have, remember, I have to put a parenthesis here, 2 raised to the 4th power. Now what's really important about exponents is and something you should always be cautious about, is you always want to evaluate exponents before you do other operations. This is something that a lot of students will forget, but always just remember PEMDAS. PEMDAS here says that we always do exponents before we do things like multiplication, division, addition, and subtraction. Exponents are always the second thing that you do. So a lot of students, what they'll do is they'll take something like this, and they'll multiply the 3 and the 2 before they've done the exponent, and they'll do something like negative 6 to the 4th power. And this is wrong. This is wrong. Don't do this. If you do this, you're going to get the wrong answer. Alright? So just be very careful. So, really, what happens is you actually sort of have to take care of the 24 first. You have to do that before you do this multiplication. So this is really like 3 negative 3 times 2 times 2 times 2 times 2. That's what 24 means. It's just 2 multiplied by itself a bunch of times. And it might be helpful to write out all the multiplication because you may not know what 24 is off the top of your head, and that's fine because you could write this out, and 2 times 2 is just equal to 4, and then 2 times 2 is just equal to 4. So, really, this is just negative 3 and then 4 times 4. And that's a little bit easier to solve because we know 4 times 4 is just 16. So in other words, your final answer, negative 3 times 16, is actually just negative 48. And that's the answer that's how you evaluate an expression with an exponent in it. Alright? Now let's take a look at the second problem here. Here we have y2 plus 102. Alright? So remember, evaluating an expression just means that I'm going to replace a y with a 5. Some other words, I just replace the y with a 5 over here, 52 plus 102. Alright? Now remember, order of operation says we have to do the exponents before we do anything else like addition or subtraction. So first, take care of the exponents. This is really just 5 times itself, 5 times 5 plus, and this is just 10 times 10. So remember, you kind of just do those two things first before you do the addition or subtraction. So in other words, the 5 times 5 is just 25. That's what this becomes. Plus the 10 times 10 is just a 100. So, therefore, your final answer is a 125. Alright? So that is the answer. Now last but not least, we have you could have expressions involving multiple exponents and even multiple variables. So let's take a look at this one here. Here, remember, x equals 2, so we just replace the x with a 2. And y equals 5, we just replace the y with a 5. So in other words, this just becomes this x to the third power. This actually just becomes 2 to the third power plus 4 times, and this just becomes y. So that's a 5, and so this is just going to be minus 7. Okay? So remember, we have to do the parentheses, and we have to do the exponents first. In other words, we have to take a look at this before we can do the addition or subtraction, and we have to do the exponents before we do the addition. So in other words, we have to take care of this 2 to the third power first before we do anything else. So in other words, what happens is 2 to the third power is really just 2 times 2 times 2 plus, and then we have 4 times negative 5. So 4 times negative 5. And then we have and then we have minus 7 on the outside here. Alright. So we have to do the multiplication before we do addition and subtraction. This actually ends up becoming 8 2 times 2 times 2, and this 4 times negative 5 actually becomes negative 20. So now because I'm doing addition and subtraction, I can actually just drop the parentheses over here. This is really just 8 minus 20. Alright? So 8 minus 20, but this is still in the parentheses, so you have to just drop that. So you you can't drop that. And then minus 7, so we have to do this thing first. And this 8 minus 20 is just negative 12. Negative 12, now we can drop the parentheses, minus 7, and this just becomes negative 19. Alright? So this whole expression here evaluates to negative 19, just plugging in a bunch of letters for numbers, and that's how to deal with exponents and expressions. Let me know if you have any questions. Let's move on to the next video.
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- 0. Review of Algebra4h 16m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations43m
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- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 6m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 19m
- 10. Combinatorics & Probability1h 45m
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Algebraic Expressions
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