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Robert McKeown: Hi, everyone. Welcome back. It's

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ALEKS walkthrough number seven, and I am your

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host, Robert J. McKeown. I'm very pleased that

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you've decided to watch this video. As I

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mentioned in in video six, being able to

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manipulate an equation is very important. As an

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economist, and as an undergraduate student in

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economics, you're going to be asked to do that

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many, many times. And today's topic is about

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radical expressions. So that's like square

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roots, square roots show up a lot. And one

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reason why square root functions show up so much

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in economics, is because they display

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diminishing marginal returns, you're going to

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hear an awful lot about marginal. Everything's

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in economics, in economics, we say, almost all

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the excitement happens at the margin. And when

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we say margin, you can kind of think of that as

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like a border. And when we have diminishing

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marginal returns, well, that's just like, you

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know, if you have one ice cream cone, you know,

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one ice cream scoop, you're going to enjoy that

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a lot. If you have a second one, maybe you'll

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enjoy that a lot, too, but maybe not as much as

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the first one. And if you have three, well,

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maybe now you're going to get a stomachache. So

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you're not going to enjoy that as much as the

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second one or as much as you enjoyed the first

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one. So that's sort of the idea. So square roots

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show up a lot in economics, and it's very

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important to be able to manipulate them.

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Fortunately, as you'll see, it's a lot like any

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exponent. Now, before we jump into the video,

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and we start doing some work together, don't

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forget, you should have your pencil, you should

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have some paper, scrap paper, a notebook, what

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have you. And if you've got the money, invest in

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a tablet, because if you write it on this

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screen, you can save it, and then you can keep

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it in an electronic file forever and ever. And

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with that, let's do some problems together.

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Let's start off talking about what a radical

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expression is. a radical expression is any

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mathematical expression that contains this

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symbol. That includes the square root but not,

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but it's not exclusive to the square root. So it

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could also be a cube root, or any root to any

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number that you like. A few definitions, many

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roots create what we call an irrational number.

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And the definition of a number that's irrational

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is one that cannot be written as a simple

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fraction. So for example, the square root of two

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and pi are irrational numbers. When working with

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square roots, you always want to remember that

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it's not possible to take the square root of a

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negative number, the square root of a negative

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number is defined. And these are sometimes known

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as imaginary numbers. imaginary numbers are

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important if you're studying quantum mechanics,

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quantum physics. But they're not important, or

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at least I've never found them to be important.

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In economics, it's not something we use in

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economics. In economics, we focus on the real

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numbers, which I'll show you. In this next

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slide. We've been asked to graph a square root

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function. So it's a very, it's the most simple

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one we can possibly have. It's the function

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which is equal to the square root of x. And this

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function will look something like this. Maybe

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I'll label it f of x. And if x were equal to

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four, and has I haven't drawn it to scale very

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well, y is going to be equal to two. And if y is

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equal to three, x must be equal to nine. Now

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everything in this space, these are real numbers

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over here, and then anything over here at least

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on the x axis, focusing on the x axis here. We

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don't know these are undefined. For our

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purposes, the square root of a negative number

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is undefined. We don't know what it is. We don't

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know what it is. If you want to learn more about

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them, by all means, take a course on complex

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numbers taken in Dance mathematics course, and

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knock yourself out.

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Here's our first ALEKS question. We're asked to

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find the domain. What has been described or

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given to us is a radical expression to radical

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expression because it has a square root. And

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we're supposed to write our answer using

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interval notation. So this is a more complicated

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square root function than the one we did

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together on the previous slide. Now, we don't

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need to draw anything, we don't have to graph

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anything, we just have to figure out for which

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values of x is there an answer for y. So rewrite

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our function as y is equal to negative x,

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negative eight x plus 24. And the first thing we

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can notice here is that if x is equal to three,

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y is equal to zero. Or, well, I'm getting ahead

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of myself. Y is equal to the square root of

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zero. And the square root of zero is just equal

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to zero. If x is greater than three, then y will

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be equal to a negative number. Which means y is

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on defined. So we won't know what y is if x is

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larger, strictly greater than three. If x is

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less than three, then y is equal to the square

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root of a positive number. And y is defined,

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it's going to be equal to y is equal to a real

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number. So by just sort of looking at the

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expression, and making sure that I'm not taking

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the square root of a negative number, I know

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what the domain of this function is. And an

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interval notation I might write x is in this

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space, well, any number less than three will

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give us a positive value for y. So I can go all

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the way to negative infinity. Let me try and

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write that again, negative infinity. And the

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number can be x can be as large as three. And

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I'm going to make sure I put a square bracket,

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because x is allowed to be equal to three or

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less than three. So here I am on ALEKS. And it's

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asking me to write my answer in interval

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notation. So I guess maybe I'll start off by

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picking the correct brackets that I want. And

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it's this one right here, where it's a rounded

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bracket on the left, because there it's an open

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open space. And it's closed on the right, it can

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only go up to three. And I'll just click this

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negative infinity symbol. And then I'll click

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into the right side box, and I'll put in the

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number three. And I'll click on Check. And we

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got the right answer. Notice it didn't have to

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do the axes in this interval, or axes in this

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set, we just had to put in the set itself.

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That's all ALEKS wanted. Let's do two things

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with this question. First, I'm going to show you

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some alternative expressions for this thing up

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here, this radical expression, and I'm going to

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then simplify it, I'll put it into a simple

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expression as possible. So there's a connection

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between square the square roots, cube root

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expressions and exponents. So I could rewrite. I

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could rewrite this as

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an exponent like that. And we already learned

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some exponent rules. So this is good for us

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because it turns out square symbols, square

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roots, cube roots, they all Follow the same

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exponent rules. And often in economics, we just

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choose to express everything and exponents.

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Because it's a little bit maybe a little bit

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easier that way. But you need to be able to go

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back and forth between exponents and square root

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symbols, effortlessly, effortlessly. And this is

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going to be really great practice for you to do

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that. So according to the exponent rules, I

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could rewrite this again as 250 to the power of

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one third, multiplied by x to the power of one

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third. And if I wanted to, I could rewrite this

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as the cube root of 250 times the cube root of

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x. So I'm still I'm using these exponent rules

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that we learned previously. Now, let's simplify.

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So let's simplify this expression up here.

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I can, there's not too much I can do with it.

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But I'll notice that if I have two times 125,

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times x, so I'm sort of factoring out that 250 a

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little bit, I can write this in a more

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simplified version by noting that five times

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five times five, which is equal to five, three

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is equal to 125. And so I can take that 125,

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outside of the square root, so I'm going to have

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to be careful with my writing, I've got the cube

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root of 125 times the cube root of two x. And

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this thing on the outside is just going to be

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equal to five. And I've got five times the cube

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root of two x. And that's as simple as I can

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make this expression. Okay, let's tackle a more

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challenging problem. You can read the expression

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in front of you, you can see it on the slides

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and the video. How are we going to go about

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simplifying this thing? Well, there's a few

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different ways to go about it, I suppose. And

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I've been sitting on the fence how I'm going to

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show it, but let's do it this way. So let's

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notice that the square root of 24 can be written

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as the square root of four times six. That would

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be one way to do it. Or maybe, yeah, okay, I

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guess that's fine. But we could go, we could go

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a little, we could go further, we could say,

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well, this is also equal to the square root of

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two times two, that's four, times two times

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three, two times three is six. So I've now

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written that in its lowest possible factors.

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Similarly, I can look at the square root of 98.

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Now while the square root of 98 has no

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particular significance for me, I know that

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anything, that's a pot, it's an even number

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divided by two is going to produce a whole

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number. So if I rewrite this as two times 49.

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I'm a little bit closer to something useful

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because then when I see 49, and I know my

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multiplication table, I know that seven times

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seven is equal to 49. So I'm going to go a

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little bit further here. I've got two times

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seven times seven. And so why don't I bring this

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all together? And Alright, I've got the square

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root of two times two times two, times three.

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multiplied by three, and multiplied by two. Why

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should be careful here? Square root of two,

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seven and seven. Now, I can do a few things. So

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I noticed that maybe I want to look at the fact

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I've got four twos, I've got the square root of

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two, four times, so I'll rewrite this as the

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square root of two to the power of four, and

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I've got two threes. And I'll rewrite that out

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note, excuse me, I've got a square root of

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three, and I've got a three itself. So I'll just

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put the three there. I'll put the square root of

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three there. And I've got two sevens. And so

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I've got the square root of seven squared. Now,

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you should see that if I've got the square root,

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and I've got the square, they're going to cancel

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each other out. So I've got two square to the

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power of four, that's going to give me four.

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And I've got the three that's still sitting

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there. And I've got the seven. And I've got the

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square root of three. And now I'm going to have

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12 times seven times the square root of 312

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times seven. Oh, my multiplication table up what

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the 12 is a little shaky, but I know that seven

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times three is 21. And I know that for 20 ones

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is equal to 84. So notice that I'm doing this

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I'm trying to do this without using a

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calculator. If you use it with a calculator,

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there wouldn't be much of a challenge to it, but

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it's handy. Now we've got our answer. Let's put

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our answer into ALEKS. So I'll start by typing

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84. And then I'll click this button right here.

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So the first little box represents the 84 and

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then I've got the square root symbol that's

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going to be put in for me, and I type in three.

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And if I hit the check button we've got the

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right answer.