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Robert McKeown: You're going to get a lot of

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questions about exponents. And the best way for

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me to help you answer these questions and

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understand different exponent rules, is just

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that to walk you actually through the rules

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themselves. So let's take a look at the first

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problem in your slides. Now, I'm just going to

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break this down and just let's look at four. Oh,

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let's look at four to the power of three. What

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is that saying? Well, it's saying four should be

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multiplied by itself three times. So that's

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four, times four, times four, and four to the

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power of two, is four should be multiplied by

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itself twice. So we have four times four. Now,

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the very long winded answer to this question

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right here, is going to be well, for the power

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of two times four, the power three is four times

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four, times four, times four. And you can see

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that that's four to the power of five. Now, why

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is that important? Because there's an easier way

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than writing out and this long form the way I

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have to get the answer. And that's shorthand if

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the bases are the same. In this case, they are

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they're both equal to four, then we can add the

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exponents, four to the power of two, plus three,

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while that's gives us four to the power of five.

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So we didn't have to go through that

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intermediate step, we can just use the shorthand

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rule to get the right answer. Now, I mentioned

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to you that four is the base, so four is the

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base to the exponent. One thing to keep in mind,

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this is when we're multiplying one base times

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another, if I wrote four to the power of two

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plus four to the power of three, that is not

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going to be equal to four to the power of five.

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Now, if you're sitting there, and you're going,

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Yeah, of course, it's not that would be stupid.

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This is a common common mistakes students make

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many, many times very, very often students make

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this mistake. Don't be one of them. Practice,

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practice. So these rules become second nature.

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And you'll never you'll never make a mistake

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like that. But if you don't practice, do you

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think you won't make that mistake, but when

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you're sitting in an exam, you can't get the

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answer, and you start to stress out and you

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start to panic. Those are the kinds of mistakes

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students make. I see it all the time. Now let's

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look at the next question. The question at the

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bottom of your slides, we've got a fraction.

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Well, if we've got a fraction raised to a power,

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it's still the same methodology, we've got three

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divided by five, multiplied by itself three

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times. So I could write that out as three over

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five, times three over five, times three over

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five. Notice that I could also write it as three

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to the power of three over five to the power of

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three. What I want to say here is that the rules

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apply, even if the base is a fraction. So don't

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be confused at the basis of fraction. Also, keep

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in mind that the factor or excuse me, the

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exponent is coming into the fraction. Now what

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are we doing see four raised to the power of

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negative one? Well, that's actually another way

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of writing one over four. So another thing, when

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I'm used to thinking of is this is the inverse

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of four, it's the inverse of four. Now what if

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we have two factors that are multiplied by each

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other with negative negative exponents? Well,

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there's a few things few different ways I could

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rewrite this one I could do is point out to you

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that we've got four raised to the power of

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negative two, well, that's like being raised to

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the power of two times negative one. And we've

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got four to the power of three times negative

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one. And so that's going to be like one over

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four to the power of two, times one over four,

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the power of three because I can bring that

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negative and switch the numerator and the

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denominator accordingly. And so another way I

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could write this, I've got one over four

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Multiply by itself How many times? Well, two

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plus three, or one over four to the power of

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five. That's what it looks like. more simply, I

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could say, well, I've got for the negative two,

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minus three, which gives me four to the negative

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five. All these different ways of writing are

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equivalent, I can add the exponents, even if

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they're negative, I can add the exponents, even

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if they're negative. What's four to the power of

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zero? You might be tempted to think that it's

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equal to zero. But it's actually equal to one.

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Why is it equal to one? Well, we know that for

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say, to the power of two minus two, well, that

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should be equal to four times four times one

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over four, times one over four, and that's going

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to be equal to one. So four raised to the power

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of zero is equal to one, it's not equal to zero,

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it's actually it's actually equal to one. Now

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what about exponents raised to another

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exponents, so we've got four squared raised to

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the power of three. Well, I can write that out,

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like four squared, times four squared, times

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four squared. And that's equal to four. You

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guessed it four, times four, times four, times

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four. And this is equal to four to the power of

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six. So if I have four squared raised to the

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power of three, that's like having four raise to

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the power of two times three, like so. And I end

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up with four to the power of six. So I don't

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have to do it this long way and write it all

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out, I can just jump to these rules. And I know

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that if I have an exponent raised to another

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exponent, they're gonna multiply each other.

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Okay, so those are some of those are the power

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rules that I wanted to emphasize to you. Now

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let's look at an example. Question from ALEKS.

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So we've got this messy bunch of exponents.

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We've got two unknown variables, an X and a Y,

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and a bunch of exponents floating around. And it

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says Write your answer without using negative

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exponents. So we have to transform every

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negative exponent into a positive exponent. How

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are we going to do that? Well, I'll start off by

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dealing with this negative three exponent. And

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so what's that going to look like? Well, every

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factor in the bracket is going to be raised to

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the power of negative three, so I've got four to

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the power of negative three, I've got x to the

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power of negative two, multiplied by negative

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three, right, we're gonna multiply the

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exponents. And I've got y to the power of three,

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multiplied by negative three. So I'm just going

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to leave four to the power of negative three as

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it is for now. Then I'm going to multiply out

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these exponents. So we've got x to the power of

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six, and we've got y to the power of negative

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nine. Now we need to get rid of the negative

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exponents. So any factor that have has a

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negative and an exponent is going to move to the

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denominator. So we're gonna have six to the

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power, or excuse me, x to the power of six, over

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four cubed times wide and the nine. Now ALEKS

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would like us to multiply out that four cubed.

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So we're going to have x to the power of six.

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Four times four is 1616 times four is 64. Y nine

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and so there's our answer. Now we can take a

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look at ALEKS. Now I've been recording this

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video and I already put the answer in ALEKS.

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I actually forgot to hit the record button so

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the answer is already in there. But you can see

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that we have the correct answer. The correct

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answer, just like we did on the slides is x to

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the power of six divided by 64 times y to the

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power of nine. Sometimes we need to factor by

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grouping. This is a more challenging way of

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factoring. But I just want to try and help you

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help you with it. It's not very difficult, as

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long as you recognize what the question wants

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you to do. So we can't This is not a quadratic.

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And since it's not a quadratic factoring, it is

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not going to be straightforward. But I want one

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thing I can do I notice here is, if I, I'm going

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to sort of why we call by grouping is I'm going

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to have two groups, we'll call this a, and we'll

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call this B. And so I'm going to focus my

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attention on a and I'm going to try and make a

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look like be good to try and make a look like B

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How am I going to do that? Well, if I just look

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at a in isolation, I can see that I could factor

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out a two and a V squared. Two is can't be

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divided into 10 or eight and get a whole number.

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If I do that, I'm left with five v plus four.

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And then over here, we have minus V minus four.

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Now notice that I've made Group A look more like

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Group B.

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Now, they might look the same, but they're not

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exactly the same yet. And one thing I can do to

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make them more similar is I could factor out a

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negative these negative sign so I could rewrite

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this whole thing as to v squared, five v plus

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four plus negative one, times five v plus four.

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Now, they're the same. So I can have to the

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square Oh, be careful. minus one, five V, plus

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four. So I've essentially factored out

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five v plus four in that bad whole bracket, this

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whole bracket, right here. And when I do that,

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I'm left with two v squared minus one. And that

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whole thing is going to be multiplied by five v

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plus four. So here we are on ALEKS. And I'm

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going to put in the answer, but two v squared.

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And then make hit the right arrow minus one Oh,

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and I forgot my brackets. So I'm going to go

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back and put brackets around that factor. And

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then another bracket, and I've got five B plus

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four. And it looks looks a little funny. But

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I've got the right answer. The brackets are

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different sizes for some reason, but I guess

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that's just the way I typed it in. And we we did

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it we factored by groups. And so I divided the

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question into a group A and Group B. And then I

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made Group A look as much like Group B as

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possible. So we're told to combine like terms

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and simplify this expression before us. So I'm

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going to start off by multiplying the factor

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into the bracket and, or what I like to call

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opening up the brackets, I'm going to open up

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the brackets, so I've got negative four times

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negative five u, so that's going to be plus 20 u

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plus four y plus 15 y minus nine you so a

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negative times a positive is a negative right.

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We saw that in a previous question. And so I've

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got those groups. I've got the U groups. And

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I've got the Y groups. So I've got two types of

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terms. And when I add them together, I'm going

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to have 11 u 20 minus nine plus 19 y. And

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typically, the term should appear in

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alphabetical order. So we've got the you have

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the Y here. Now let's put it into ALEKS and see

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if we've got the right answer. So here we are in

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ALEKS, we had 11 u plus 19 y. And let's see what

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happens when we hit the check. We had the right

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answer, we collected the terms successfully.