WEBVTT

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So what we consider next is a fractional power.

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Fractional power.

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And what is happening?

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Okay, wait.

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Okay.

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Okay.

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So we're looking at the problem where the power of the in the AC is in the form of a

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fraction.

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Because it doesn't meaning we define it and it, it, it can be, can be work out from,

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from this example.

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So if I will consider 25 to the power of half.

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Okay.

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And all of that for the power of two, then according to the properties which we define

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before is we have to multiply those two numbers together.

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Okay.

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So half I have a two and the properties are telling me you need to multiply those things

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together.

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So if I do that, I will have a 25.

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So that will then raise the question, what sort of object is in this bracket before I

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square it?

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If I square it, I will have a 25.

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So what do I need to square to have a 25?

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And the answer is there.

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You're looking for the square root.

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So the fractional power one over two or the fractional power half means square root.

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So square root over 25, it's a plus and minus five.

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So that's fine.

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Square root is perfect.

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And in a similar way, we can deduce the meaning of these fractional powers by considering

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the numbers, the, the inverse power.

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So if I will look at the minus one, two, five to the power of a third, I take all of that

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to the power of three.

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So that will be minus one, two, five to the power of one.

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And anything to the power of one, is that anything?

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Nothing changes.

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So that's a minus one, two, five.

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So the question comes again in the same way as we did with the square root, you know,

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what is the number which I cube and I will get the minus one, two, five.

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And if you do a little bit of a thinking, you should get or deduce that it is equals

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to minus five.

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Okay.

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Everybody following?

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Yeah.

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Good.

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So in general, this is what we want in your list of important properties and formulas.

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If you're looking at the x to the power of one over n, it's the nth root.

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Nth root of whatever you're looking at as your x.

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Okay.

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So it's the nth root.

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Given this definition or given this way, we write those things, everything just follows.

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And hence, I don't know, four to the power of half is a square root of four, or better

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to say the second root of four.

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That's a plus or minus two.

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Similarly with this one.

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What is the eight to the power of one over three equals to eight to the power of one over

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three.

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So according to our way to write those things or the property, the law, we can say that this

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is a cube root of eight.

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A little bit of a thinking, well, this is the cube root of eight.

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Eight to the power of five.

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Eight to the power of five.

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This here.

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No, I don't follow.

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Say again.

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Number one, I.

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Eight to the power of five.

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Eight to the power of five.

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Oh, you mean my writing?

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Oh, yes, yes, yes, yes, yes.

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Yes.

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This is square root of four.

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This will be square root because it's the two which refers to the root.

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What we don't do is when you have a square root, you don't put the two in there.

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So if you will have x to the power of one over five, that is the fifth root of x.

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If it's an extra power of five over five, then it's the fifth root of x to the power

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of five.

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But before I will say that, I will say, oh, wait a minute, extra to the power of five

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over five is extra to the power of one, which we just defined.

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It's the same thing.

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Yeah, I will first cancel it.

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Say this is just x to the one of we go.

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Okay, so anybody wants the cube root of eight?

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Shout at me.

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Two, two, anybody else with two?

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That is equals to two.

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Why is it not minus two?

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Well because x, if it's minus two, then I have a minus two times minus two times minus

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two.

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So that is equals to a minus two times minus two is four, which is cool.

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But I have to multiply it by minus two, which will give me minus eight.

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And minus eight does not equals to eight, which is what I am after.

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Okay.

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Now there is a funny thing.

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And we are looking into the future when you will be like masters of the complex numbers

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and you can do all this imaginary stuff.

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If you're looking at these roots of those numbers, then whatever is the root, like the

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nth root, so many numbers you will get.

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Okay.

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But what we don't see is that there are another two roots.

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There are three roots to this problem, but two of them are complex.

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And we first need to get our hang of complex numbers.

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So in this stage, you just do your thinking and you may use your calculators and try a

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couple of things to see whether or not that is doable.

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The other thing you can do, and that will be interesting, is to go on to your fancy

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calculator if it's going to load up and see what's the fancy calculator is going to say.

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Yeah.

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So what was it?

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Qubit root of eight.

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So it's eight to the power of, this was one over three, wasn't it?

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One over three.

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Eight to the power of one over three.

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And it tells me two.

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You see?

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My calculator also says that check your little calculators if they do it as well.

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It will be a moment.

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This is the whole point.

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There will be a moment in the near future where we'll be doing some example.

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And I will show you that this fancy calculator does give you the answer and the little calculator

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doesn't.

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Okay.

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Cool.

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So let's go.

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A couple of more here.

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I will skip them to consider.

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And I like the example of 1.5 where we will start to combine those things together.

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So we're looking essentially at the different notation of the different powers.

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So this is a square root of three.

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So there is an invisible two here.

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I said if it's a square root, we don't really need to write it.

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So it's like it's there, but we don't see it.

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And the reason why I say it's there, because I can then write it as x to the power of three.

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And that all is to the power of half.

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Yeah.

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That's the square root.

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That's what the square root is.

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You take the mathematical object and you take the power of it to the half.

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That's a square root.

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But now it is in the form which we can use and apply the properties of the powers we

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learned at the very beginning.

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This is a x to the power of a, all of it to the power of b.

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And according to the property which we defined at the beginning, we can write this as x to

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 the power of three times one over two.

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So this is x to the power of three over two.

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And I wouldn't take it anywhere further.

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That's simple enough for me.

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So it's x to the power of three halves.

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Here this is a fifth root.

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Fifth root of x squared.

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So I have this object x squared and I'm taking the fifth root of it.

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Well, fifth root in general, we can write it as well, the object is x squared and the

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fifth root is one over five.

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So it's the power to the one over five.

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Okay.

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But now I'm home and dry because we have it in the form as we can apply these laws and

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properties from the beginning of the class.

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So we can say this is x to the power of two times one over five, which is x to the power

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of two over five.

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Yeah.

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Everybody following?

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Yeah, some nodding there, some nodding there.

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Yeah.

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Yeah.

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Yeah.

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Cool, cool.

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As I said, if you don't tell me, if you shy, wait until we finish and tell me anyway,

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we need to be able to say those.

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So in general, here is the formula, here is the important result of this whatever 10, 15

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discussion that if I have a expression or a term in the form of end root of M, okay,

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oh, there is a typo, sorry, that should be an end root of X to the power of M. Yeah.

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There is a typo.

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I need to change it all.

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So it's end root of X to the power of M. Then I can write it as X to the power of M divided

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by N, okay.

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And the other general way of writing things, if I have a square root of A and B as an object,

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I can split them up and I can write it as a square root of A into square root of B. And

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vice versa, both of those things works both way.

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There is an equal sign in between them.

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Those things on the left hand side and right hand side are identical.

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Okay.

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So if I look at this example 1.6, I have a square root of X divided by X to the power

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of 2 times X to the power of half.

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Now I can approach it more than one ways.

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And this is where the uniqueness of people comes in because I may do it first, I consider

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the denominator and then I consider the denominator with the numerator or I just consider all

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of them at once and write them out in the terms.

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So what I could do is I can just write, okay, look, square root of X, that's X to the half.

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Yeah.

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One over X squared, that's X to the minus 2, so that's X to the half times X to the

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minus 2.

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And one over X to the half is X to the minus half, so that's times X to the minus half.

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Okay.

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X to the minus half.

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And now I use the rule that if I'm multiplying those same terms with different powers, then

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I will just add up those powers.

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So this is equals to X to the power of half minus 2 minus half.

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This guy goes with that guy and I am left with X to the minus 2, which I can write as

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X, one over X to the power of 2.

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If I want to, these two things, I wouldn't say one is more simple than the other.

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Yeah, so if anybody is asking you to simplify or answer as much as possible, these are equal,

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at least to me.

201
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So that's how I did it.

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The other way to deal with this example will be to say, oh, I do it in two ways.

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You know, I rewrite the numerator as it was, so X to the power of half, but I will first

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deal with the denominator.

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So I will write it as X to the power of 2 plus a half.

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Yeah.

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So that is then equals to X to the power of half divided by X to the power of 3 halves.

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So I did like intermediate step to simplify it a little bit and now I am looking at two

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terms.

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So I can do what we have listed in the properties.

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So this is X to the power of half plus X to the power of half minus 3 over 2.

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Yeah.

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And the surprise, I will get the same answer.

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This is, wait, what?

215
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Oh, that's not three halves, is it?

216
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Why are you not shouting?

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It's five.

218
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Who said that?

219
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Thank you.

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That's the moment when you will be like, yeah, he told us to be quiet for two hours, so you

221
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should be like waiting for these things.

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And go, wow, what have you done?

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How could you?

224
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So it's a five.

225
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No, this is still one here.

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Five halves.

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Yeah, it's here and there.

228
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So five, five.

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So this is X to the power of minus 4 over 2, which is X to the power of minus 2.

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Same as before.

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Okay.