Common Core Algebra II.Unit 1.Lesson 3.Common Algebraic Expressions

Subtraction multiplication and division, along with exponents and roots, right? So we have simple algebraic expressions, like three X plus 7, slightly more complicated ones, 5 X minus two divided by X plus four. Then things involving maybe a square root, like four times the square root of X -5, plus three. And absolute value of 7 minus three X divided by two X plus 8. Now with all of these for a lot of math people, math teachers, people that use math, physicists engineers. They look at these kind of expressions and it's just obvious sort of the order of operations. In other words, how you would evaluate them if you were given a value of X for students even at the algebra two level though, this may not be automatic at this point. So that's what we're going to be working on a lot today. Evaluating sort of more complicated algebraic expressions. So let's get into it. Exercise one, consider the algebraic expression four X squared plus one. Let a race is described the operations occurring within this expression and the order in which they occur. Pause the video now and think about that for a moment. 

All right. Well, being able to quote read an expression like this is very important. You know, it's tricky for me because I almost feel like I give it away as I read the expression. But the plain fact is the first thing that we're doing is we're squaring X, right? Because those exponents come before everything else. After squaring the X were then multiplying. By the number four, right? Because multiplication comes before addition. And the final thing that we're doing is that we're adding one. So when I look at this expression, I can't think that I'm doing X and squaring it, then adding one, then multiplying by four. I can't think that I'm doing that, right? I have to look at this and go, yeah, I took X I squared it. I multiplied it by four. Then I added one. Now let her be says evaluate this expression for the replacement value X equals negative three. Show each step in your calculation. Do not use a calculator. Again, we've talked about that. So take a minute and figure out what the value of this expression is. All right, let's go through it. Well, again, just kind of showing my substitution and keeping in mind my order of operations, right? Just from letter a, I've got a square that first, negative three times negative three is positive 9. I then have to do the multiplication, which is four times 9 gives me 36, and then I do the addition, and I get 37. So not too bad, right? And not the most complicated algebraic expression. 

But still, one that we would like to be able to evaluate by hand. All right? Pause the video now and then I'm going to clear out the text. Okay, here we go. Let's move on. Exercise two, a little more complicated, a little more complicated. Consider the more complex algebraic expression known as a rational expression, four X plus three divided by X cubed or X to the third -7. Letter a says, without using your calculator, find the value of this expression when X is equal to three. Reduce your answer to simplest terms show your steps. See if you're ready for this. Pause the video now and see if you can figure out what the value of this expression is when X is equal to three again. Don't use your calculator. All right, let's do it. Well, it's kind of interesting here because, you know, we can certainly put that three wherever there's an X, there's no question about that. That's easy enough. Good idea wherever there's an X to always kind of put parentheses in. But now the question is, well, what order do I evaluate this in? The answer is there's a lot of different orders you can go in, but at the end of the day, the division should really come last. And by the division, I mean this big division here. 

In other words, there's always sort of an implied parentheses going on in both numerator and denominator. So if I look at that numerator, I have to do the multiplication first. That's going to be four plus three, which is 12. I'll leave it that way right now. The denominator I've got to do the three cubed three times three is 9. Times three is 27. Right? Now, again, I know division comes before addition and subtraction in order of operations. But again, this idea that I have these implied parentheses in the numerator. So 12 plus three is 15. 27 -7 is 20. And then obviously we can reduce that by dividing both numerator and denominator by 5. So my final answer is three fourths. Now exercise B is kind of important too, because I know that a lot of students will want to do this on their calculators. And generally speaking, that's okay, as long as you really still understand the math. Letter B says if a student entered the following expression into their calculator, it would give them the incorrect answer. Why? So let's take a look at this. The student enters in this expression. And it's not going to give them three fourths at to think for a while what it would give them. It'd be messy. It'd be all kind of like decimal. And the reason that it wouldn't work correct is because in terms of order of operations, this is how the calculator would interpret it.

It would say, well, I've got four times three. And it would, oh, oh yeah, I've got to do the exponent first. So it would be three to the third. Then it would look at this and then say, I've got to do the multiplication. Then I've got to do this division. Now, three divided by 27 is one 9th. And then we get this. And then it would just add these up. And again, it would be kind of ugly because of the one 9th. The reason that it's not going to give us the three fourths in any way, shape, or form, is because it doesn't know, it doesn't know that the numerator and the denominator have to be enclosed in parentheses. You would have to do that for it. Calculators are generally not that smart. Now, many calculators these days have operating systems that when you do a division immediately puts a fraction bar up there. You'll see how many calculators you'll get something that kind of looks like this. In which case those parentheses aren't necessary anymore, right? We used to call this pretty print in the business because it just looks the way that you want it to. All right, well, more on calculator use later. 

That's not the point of today's lesson. Pause the video now. And then we'll move on. Okay, here we go. Let's make these. Even uglier. Exercise number three. Is the absolute value expression? Absolute value X -8 plus two. Equivalent to absolute value of X plus ten. How can you check this? Well, remember, two expressions are going to be equivalent, right? If they have the same value. For every X for every X every X so the way that we can test that is by grabbing a value of X and trying it, seeing if they're the same. Now remember, absolute value R is all about positive and negative. So I don't know that I'd really want to grab a positive value of X, I could. But let me test X equals negative 5. Okay, I'm going to test this. All right. Now this is a very important because some students might look at this and go, oh, sure, you know, I'll take the negative 8, I'll turn it positive, I'll add it to the two. And I'll get positive ten. But let's actually kind of test what would happen if I put negative 5 into this expression, right? So I'm putting it into the expression X -8, absolute value plus two. Well, just like before, the way absolute value works is it kind of assumes that you're going to do everything inside of the absolute value bars before doing the absolute value. So in other words, I don't do the absolute value of negative 5 in the absolute value of negative 8. I do negative 5 -8, and I get negative 13. 

Then I find it's absolute value, and that's positive 13. And then I add that to two, and I get 15. On the other hand, let's put X equals negative 5 into. This expression. All right. Well, when I put negative 5 in. And take its absolute value, I get 5 plus ten. And I get 15. Now boy, that gives us some very, very convincing evidence. Very convincing evidence that these two are equivalent. And yet, they're not. This seems to imply seems to imply they are continuous. But what happens if I took something like X equals, I don't know, let me go with X equals two. Let's take a look at X equals two. If I put it into this expression, let's take a look what I have. Two -80. Gotta watch out. That looks like a 12. Now remember I had to do what's inside here first, two -8 is negative 6. And then the absolute value of negative 6 is 6, so I get 8. Let's now put X equals two and this one. Well, the absolute value of positive two is positive two. Uh oh. So not equivalent. Now you might say, yeah, but they were equivalent for the last one. Except equivalent expressions have to be equivalent for every value of X so there's a little bit of a danger in testing one value of X and saying, that's good. It's good. We're all set. You really need to test maybe a few of them. All right, pause the video now, and then I'm going to get rid of the text. Okay, here we go. Let's take a look at exercise number four. Now we have a square root involved. 

Consider the algebraic expression 25 minus X squared square root, which contains a square root. Nice. Evaluate this expression for X equals negative three. Let me ask, why can you not evaluate the expression for X equals 13? Well, first, take a moment and try to evaluate the expression for X equals negative three. You should be able to do this without your calculator. All right? All right. Well, let's take a look. We're going to see a very, very common theme on a lot of these things, which is that when we have these higher order operations like square roots, really, there's an implied parenthesis underneath them. In other words, I've got to do what's underneath them first. Now, when I look at what's underneath them, 25 minus negative three squared, now my normal order of operations kick in. Negative three squared is 9. And then when I do 25 -9, I get 16 and remember the square root of 16 is positive four. Not plus or minus four. We're only going to get that plus minus issue going on. When we solve equations involving X squared, and we're not solving an equation here involving X squared, we're just evaluating an expression. But we get four. Letter B, why can you not evaluate the expression for X equals 13? Go for it. See what happens. Well, let's take a look. Again, I'm going to throw in that implied parentheses. 

Now I know 13 is a fairly large number. But 13 squared is one 69. Now I have to be very careful when I do 25 minus one 69. I get negative. One 44. But I can't do anything with that because at this point at least I can't take square roots. Of negative numbers, right? And again, I hope that you all understand why that is. I mean, a logical guess here would be 12, but 12 times 12 isn't negative one 44. Another logical guess would be negative 12, but negative 12 times negative 12 isn't negative one 44. So we can't take square roots of negative numbers. At least for now, wait until later in the course. All right, let her see. It's about time I brought one of my kids in. Max thinks the square root operation distributes over subtraction. In other words, he believes the following equation is an identity. The square root of 25 minus X squared is 5 minus X that would make sense. You know, square root of 25 5 square root of X squared is definitely X, so why not? Well, it says show that this is not an identity. All right. Well, we can do that we can do that. By actually just picking a value of X substituting it into both sides and seeing if they're equal. One great choice would be to test X equals negative three. Up from letter a, right? Because we already know what we're going to get when we plug negative three into the left hand side. 

The only thing we haven't done is tested the negative three in the right hand side. Real quick, I'm going to go back through this 25 -9, I'll just leave that as 5 minus negative three for right now. That's going to be the square root of 16. Again, leave it as 5 minus negative three. Square root of 16 is four. And now remember when you subtract a negative, that is the same as adding its opposite opposite, so no. If this weren't identity, we would always get things like four equals four, 8 equals 8, 5 equals 5, but getting four equals 8 means that no, I'm sorry, those two expressions are not equal and when X is negative three. And an identity is going to be equal no matter what value of X you put it. All right. Pause in the video, write down what you need to. All right, let's clear out the text. And let's take a look at one more. This one's a beast. We have a square root, a squaring and absolute value, everything. Right? What's interesting is letter age is simply says, what operation comes last in this expression? So if I grab the value of X and I substitute it in, which is what we're going to do in letter B what would be the absolute last thing that would happen. To figure it out. Square root is last. The square root is last. All this other stuff is going on inside the square root, then the last thing that we're going to do is take the square root of that pure ugliness underneath it. Let her be says evaluate the expression for X equals two, simplify it completely. All right, you can do this. 

Take your time, put X equals two in, work it down, think about order of operations. Think about that absolute value, try to do it without your calculator, okay? All right, let's go through it. No problem for algebra two students just need some space there. Again, those are absolute value bars, not the numbers 12 and 81. And I just substitute everything in there. All right. Now, what order do we work with? Well, we can just work the numerator and denominator separately in the numerator. We have the absolute value of negative 6. Let's work the denominator. Of course, in the denominator, we've got to do that squaring first. So I'll do two times two and get four. Take as many steps as you need, the absolute value of 6 is 6. That means the numerator is done. Then in the denominator, we're going to have 20 plus four. And finally, we'll have 6 divided by 24. Yeah, no need to take the square root yet. Let's simplify that by dividing actually numerator and denominator by 6, and we get the square root of one fourth. A square root of one fourth is one half. Remember that when we multiply two fractions, we multiply their tops, we multiply their bottoms. So that's it. 

One half. All right, a lot of evaluation there, but as long as you know your order of operations and you're sort of implied parentheses in the numerator and the denominator and under the square root. And you work through it, and it comes out real nice. All right, pause the video and then I'm going to clear out the text. Okay, here we go. Last question is a little multiple choice question. It's always good to get a little bit of test prep here and there with our multiple choice questions. See if you can find their correct choice. All right. By the way, especially at the algebra two and above level, never assume that you can do a multiple choice question by just looking at the four answers and figuring out which one looks the best. Sometimes you can definitely eliminate some answers. There are some rare occasion when you can actually do that, but most of the time you can. So let's throw that ten in. All right, I've got absolute value of the square root of four times ten plus 9 minus ten squared, all divided by three. So many choices so little time. Well, I can certainly work within that square root, right? There's parentheses sort of implied in there. 

So I have four times ten, which is 40, plus 9, which is 49. So I'm going to take a little shortcut there. I can also work with this term ten times ten is obviously 100. And then I've all got it divided by three. Okay. After that, I can then take the square root of 49. And what I'll find is 7 -100. All divided by three now it gets a little tricky. Now I've got to do what's inside of that parentheses. That's going to be negative 93. Divided by three. But now, the absolute value of negative 93 is a positive 93. Divided by three. And that ends up being choice one, right? Choice one. Bubble in the dot. All right. So let's clear out this text, pause the video now. Okay, let's finish up. So in today's lesson, we looked at common algebraic expressions. In other words, things that you're just going to see throughout this year, things like absolute values, roots, and exponents. That you should know how to evaluate at this point. Now sometimes it's helpful with the calculator just to sort of check your arithmetic. But what's most important is that you're able to read an expression. You really understand what things are happening to the variable and in what order. All right? So we're going to be using these things a lot this year. For now though, let me just thank you for joining me for another common core algebra two lesson by E math instruction. My name is Kirk Weiler. And until next time, keep thinking and keep solving problems.