Common Core Algebra II.Unit 4.Lesson 1.Integer Exponents
Algebra 2
Learning the Common Core Algebra II.Unit 4. Lesson 1. Integer Exponents by eMathInstructions
Hello, I'm Kirk weiler and this is common core algebra two by E math instruction. Today, we'll be working on unit four lesson number one on integer exponents. Now you've been dealing with exponents for quite some time. But in common core algebra two, we're going to be pushing exponents to the extreme. So we want to take a few lessons at the beginning of the unit number four, which is all about exponential functions. To really talk about and review some basics about exponents. So today we're going to be looking at integer exponents and reminding ourselves of what they mean. So let's get right in them. In exercise one, what we're going to do is we're going to look at a pattern. It says the following sequence shows powers of three by repeatedly multiplying by three, fill in the missing blanks. Well, this is kind of simple enough, right? Three to the first is just three. Three, the second is three in the first times three, so it's 9. Three to the third would be 9 times three, so of course it'd be 27. But what does three to the zero mean? Well, whatever three to the zero must mean when you multiply it by three, well, you got to get three. So three to the zero must be equal to one.
Now, if we're multiplying by three to go in this direction, then we have to kind of be dividing by three, and if you think about this, we must be dividing by three to go in this direction. But that means three to the negative one, well, that would have to be one divided by three. Three to the negative two, well, that would be one divided by three divided by three, or one divided by three squared, which would be one 9th. Three to the negative three would be one divided by three cubed or one 27th. So what's important here is that positive exponents. Positive exponents, well, well, what they mean is they mean to multiply. But negative exponents. Well, they mean to divide. All right. And specifically to divide the number one, right? So three to the negative 7th would be one divided by three to the 7th. Now three to the 7th is huge number, so I'm not going to even. Go there. But that's simply what it means. It means to take the number one and divide by three to the set three, 7 times. So three times three times three times three. So negative exponents are a little bit weird, but at least they mean division positive exponents mean multiplication.
An exponent of zero, that's very odd, right? Because that tells us that we're really not multiplying anything at all, but specifically we're getting the number one. Okay? So write this down if you need to, and then let's summarize it in a table, okay? All right, clear it out the text. Let's summarize. So what we should feel comfortable with before we move on is that if N is any positive integer, all right, so 5, 17, 12, et cetera. Not fractional, not negative, positive integer. Then B to the N is just one times B N times. All right? B to the negative N is one divided by B to the N and B to the zero is equal to one. Please note that zero to the zero, well, that's what's called indeterminate. We can't do that. But any other number raised to the zero power is equal to one. So 5 to the zero is equal to one negative 7 to the zero is equal to one, et cetera. Okay. So let's get a little workout on this, okay? Let's go right to an exponential function. Given the exponential function, F of X equals 20 times two to the X, evaluate each of the following without using your calculator, show the calculations that lead to your final answer.
Now I think it's good for you to take a shot at this by yourself to see if you remember order of operations. Don't use your calculator, all right? All you're doing is robbing yourself if you do that. So pause the video now and try to work out a, B, and C on your own. All right, let's go through it. Well, obviously, you need no reminder, hopefully about what function notation is. But all I'm going to do is replace X with two. Now remember order of operations, right? We have to do this exponent before we do the multiplication. So that 20 is just going to hang out and obviously two squared is four. Now we can do the multiplication 20 times four is 80. So F of two is 80. Let's do F of zero. F of zero, same song second verse. We have to put the zero in there. Now, be careful. That's got to come first. So we don't do 20 times two and get 40. We do two to the zero, and we get one. Then we do 20 times one, and we get 20. Finally, if we do F of negative two, we're going to get 20 times two to the negative two.
Now what this really means, honestly, is that we're going to divide 20 by two twice. But you could also interpret it as follows. You could just say, well, all right, two to the negative two is going to be one over two squared. So that's going to be 20 times one fourth, and of course that's the same as dividing by four. And we get 5. Now to take this up to a little bit of a higher level, a common core level. Take a look at letter D it says when X increases by three by what? Factor does Y increase. In other words, does it double? Does it triple? Does it go up by a factor of 5? What factor? So what do we multiply by? All right, I want you to think about this a little bit. If I increased X by three, what would I multiply by? All right, well, let's take a look. Now, one thing that you could do is you could do something like this. You could say, well, I'm going to figure out what F of one is. And F of one would be 20 times two to the first, which would be 40. Then you could say, well, let me increase that by three, X by three, and that would be F of four. So then I'd have 20 times two to the fourth, and that would be 20 times, well, that would be two to the fourth, 16. And that would be 320. So by what factor did I go up? Well, I multiplied by 8. Now, why is that? Well, that's two to the third, right? Two to the third, two times two times two is 8.
So when I increased by three, it allowed me to multiply my answer by two, three more times. I got to multiply that 20 by two, three additional times over what I did before. And because of that, it goes up by a factor of 8. And it doesn't matter what we start with. It will always go up by a factor of 8. All right. Pause the video now. And write down anything you need to before we move on. All right, let's do it. Turn it up. We're just going to get a lot of work with integer exponents today. Now, one of the things that we want to do is start getting into what are called exponent properties. Some people call them exponent rules. Some people call them exponent laws. You take your pick. Exponents play by certain rules, right? And we want to know the rules of the game. So exercise three says for each of the following write the product as a single exponential expression. Write a and B as extended products first. If necessary. All right. In other words, two to the third times two to the fourth, I want a single exponential expression right now. I have two of them.
One, two. Now, of course, two to the third is just two times two times two. And two to the fourth is two times two times two times two. So obviously what we have here now is two to the 7th. And that is my single exponential expression. Why don't you try it for B and then try to generalize it in letter C. All right. Well, we could write out the same extended product, but again, here we've got two multiplying itself 6 times and here multiplying itself two times. So when we multiply those two together, we get two to the 8th. We don't get 40 eighths. Because all we're really talking about is how many times we're multiplying by two and we're multiplying by two, 8 times. So finally, we can generalize this if we have two to the M times two to the N, we'll get two to the M plus N this is probably the most important exponent law. When you multiply two quantities that have the same base the same number or variable down here, you then add their exponents. So if I had X to the tenth times X to the 5th, I get X to the 15th. If I add ten to the 7th times ten to the second, I'd have ten to the 9th. If I had X to the third times Y to the fourth, I'd have X to the third times Y to the fourth. See, there's no way to combine these two because these aren't the same. So make sure that you understand how the rule law, property, whatever you want to call it works.
All right, pause the video now. All right, and now let's get rid of some text. All right, let's keep going. Now, in that last problem, we were looking at that first exponent law. Using only positive exponents. Let's see if it works for negative exponents. Okay? It says consider now the product of two to the third times to the negative one. It says use the exponent law in exercise three C to write this as a single exponential expression. Well, we had this law that said, as long as the basis of the same, and they are, we can add the exponents. So we would have two to the three plus negative one. Which would be two to the second. Now, of course, two to the seconds four, but it just says a single exponential expression. So I'm just going to leave it like that. Letter B says evaluate but let me put the equals four down here. Let her be says evaluate two to the third times two to the negative first by first rewriting to the third and to the negative one. And then simplifying. Well, two to the third is two times two times two. So that's 8. Two to the negative one by definition is one divided by two to the first, so that's one half. So two to the third times two to the negative one. Plus 8 times one half. Which is equal to four. Hey, look at that. Do your answers from a and B support the extension of the addition property of exponents to negative powers as well. And the answer is yes. Because the results. Of a and B. Were equivalent. All right.
What we want is we want exponent properties. That are consistent. We don't want to have one exponent property that works with positive exponents, but not negative. Or works when the exponent is positive or negative, but doesn't work when the exponent is zero. All right? As soon as we have to create all these various cases, what a headache, right? But it looks like the addition property of exponents work, whether the exponents are positive. Or negative. It'll also work at the exponent zero. All right, pause the video now if you need to, and then we'll move on. All right, let's clear out the text. Exercise 5. We're going to look at one more exponent property today. And this one's going to get at it. It says for each of the following, write the exponential expression in the form of three to the X all right, we're X could be a number. It could be an expression. A and B is extended products first. If necessary. So let's take a look at this. This looks frightening, but at the end of the day, what does it really mean? We've got something cubed, right? So that means we've got something. Times that thing. Times that thing. Right? That's all we have. Three, the second three, the second three of the second. Of course, three to the second, whoops, is three times three. Then we have another three times three. And we have another three times three. But that just means we have three to the 6th.
That could be confusing, couldn't it? It almost looks like we should have three to the 5th, right? Oh yeah, the exponents. What? Not in that case. Why don't you work out letter B and then see if you can generalize to letter C. All right. Well, hopefully, you found the same pattern, which is that letter B, that's just three to the fourth times three to the fourth, right? Because we're just squaring it. So just gets written down twice, but then that's so. I won't even extend the product. That's three to the 8th. So notice in both cases, when we have something raised to an exponent raised to another exponent, we multiply the two exponents together. And that can be generalized. If we had X to the second to the 7th, we'd have X to the 14th. If we had 5 to the third to the fourth, we'd have 5 to the 12th. So whenever we have a quantity raised to an exponent, and then it's raised to another exponent, we get to multiply the exponents together. That is the second property of exponents, or third, or whatever. It's the second one we saw today. All right. Pause the video, and then we'll try to extend this property as well. All right, here we go.
All right, does it actually extend negative exponents again? Let's hope because we want that consistency. So it says consider the expression of three to the negative two raised to the fourth. Show that this expression is in fact equivalent to three to the negative 8th. In other words, that we can multiply the two exponents by first rewriting three to the minus two infraction form. So let's see. Three to the minus two is one divided by three squared. So that's one 9th. So that means three to the minus two, so the fourth is equal to one 9th to the fourth, or better yet, let me write it this way. Let me write it as one over three squared to the fourth. So that's going to be one over three squared. Times one over three squared. Times one over three squared. Times one over three squared. Now, little review when you multiply fractions, you multiply their numerators. One times one times one times one is one. But then if I do three to the second times three, the second times three to the second times three to the second, walk at three to the 8th. But that is three to the negative 8.
So it does look like that property, which says when we have something raised to an exponent raised to another exponent, we simply multiply the exponents, works with negatives as well. So if I had something like 5 to the negative second to the negative fourth, I'd have 5 to the positive 8. If I had four to the negative third to the positive 5th, I'd get four to the negative 15th. All right, so it's a pretty easy rule, really, at the end of the day. All right, when we have something raised to an exponent raised to another exponent, we multiply the two exponents. But boy, will this be important for us as we move forward? All right. Copy down whatever you need to. And then let's finish up this lesson. All right. Let's do it. So in today's lesson, what we did was we reviewed the basic notion of positive negative and zero in a direct exponent, very important to have those basics down.
Then we saw two additional exponent properties, right? When we multiply two exponential expressions with the same base, we add their exponents. And when we have a single exponential expression, raised to yet another exponent, we can multiply the two exponents. Those two properties, along with all of the basic definitions of positive integer and negative integer exponents, are critical to master and be fluent with before we move on to new lessons. So work on them. For now, though, I want to thank you for joining me. For another common core algebra two lesson. My name is Kirk weiler. And until next time, keep thinking. And keep solving problems.