So what we consider next is a fractional power.
Fractional power.
And what is happening?
Okay, wait.
Okay.
So we're looking at the problem where the power of the in the AC is in the form of a
fraction.
Because it doesn't meaning we define it and it, it, it can be, can be work out from,
from this example.
So if I will consider 25 to the power of half.
Okay.
And all of that for the power of two, then according to the properties which we define
before is we have to multiply those two numbers together.
Okay.
So half I have a two and the properties are telling me you need to multiply those things
together.
So if I do that, I will have a 25.
So that will then raise the question, what sort of object is in this bracket before I
square it?
If I square it, I will have a 25.
So what do I need to square to have a 25?
And the answer is there.
You're looking for the square root.
So the fractional power one over two or the fractional power half means square root.
So square root over 25, it's a plus and minus five.
So that's fine.
Square root is perfect.
And in a similar way, we can deduce the meaning of these fractional powers by considering
the numbers, the, the inverse power.
So if I will look at the minus one, two, five to the power of a third, I take all of that
to the power of three.
So that will be minus one, two, five to the power of one.
And anything to the power of one, is that anything?
Nothing changes.
So that's a minus one, two, five.
So the question comes again in the same way as we did with the square root, you know,
what is the number which I cube and I will get the minus one, two, five.
And if you do a little bit of a thinking, you should get or deduce that it is equals
to minus five.
Okay.
Everybody following?
Yeah.
Good.
So in general, this is what we want in your list of important properties and formulas.
If you're looking at the x to the power of one over n, it's the nth root.
Nth root of whatever you're looking at as your x.
Okay.
So it's the nth root.
Given this definition or given this way, we write those things, everything just follows.
And hence, I don't know, four to the power of half is a square root of four, or better
to say the second root of four.
That's a plus or minus two.
Similarly with this one.
What is the eight to the power of one over three equals to eight to the power of one over
three.
So according to our way to write those things or the property, the law, we can say that this
is a cube root of eight.
A little bit of a thinking, well, this is the cube root of eight.
Eight to the power of five.
This here.
No, I don't follow.
Say again.
Number one, I.
Eight to the power of five.
Oh, you mean my writing?
Oh, yes, yes, yes, yes, yes.
Yes.
This is square root of four.
This will be square root because it's the two which refers to the root.
What we don't do is when you have a square root, you don't put the two in there.
So if you will have x to the power of one over five, that is the fifth root of x.
If it's an extra power of five over five, then it's the fifth root of x to the power
of five.
But before I will say that, I will say, oh, wait a minute, extra to the power of five
over five is extra to the power of one, which we just defined.
It's the same thing.
Yeah, I will first cancel it.
Say this is just x to the one of we go.
Okay, so anybody wants the cube root of eight?
Shout at me.
Two, two, anybody else with two?
That is equals to two.
Why is it not minus two?
Well because x, if it's minus two, then I have a minus two times minus two times minus
two.
So that is equals to a minus two times minus two is four, which is cool.
But I have to multiply it by minus two, which will give me minus eight.
And minus eight does not equals to eight, which is what I am after.
Okay.
Now there is a funny thing.
And we are looking into the future when you will be like masters of the complex numbers
and you can do all this imaginary stuff.
If you're looking at these roots of those numbers, then whatever is the root, like the
nth root, so many numbers you will get.
Okay.
But what we don't see is that there are another two roots.
There are three roots to this problem, but two of them are complex.
And we first need to get our hang of complex numbers.
So in this stage, you just do your thinking and you may use your calculators and try a
couple of things to see whether or not that is doable.
The other thing you can do, and that will be interesting, is to go on to your fancy
calculator if it's going to load up and see what's the fancy calculator is going to say.
Yeah.
So what was it?
Qubit root of eight.
So it's eight to the power of, this was one over three, wasn't it?
One over three.
Eight to the power of one over three.
And it tells me two.
You see?
My calculator also says that check your little calculators if they do it as well.
It will be a moment.
This is the whole point.
There will be a moment in the near future where we'll be doing some example.
And I will show you that this fancy calculator does give you the answer and the little calculator
doesn't.
Okay.
Cool.
So let's go.
A couple of more here.
I will skip them to consider.
And I like the example of 1.5 where we will start to combine those things together.
So we're looking essentially at the different notation of the different powers.
So this is a square root of three.
So there is an invisible two here.
I said if it's a square root, we don't really need to write it.
So it's like it's there, but we don't see it.
And the reason why I say it's there, because I can then write it as x to the power of three.
And that all is to the power of half.
Yeah.
That's the square root.
That's what the square root is.
You take the mathematical object and you take the power of it to the half.
That's a square root.
But now it is in the form which we can use and apply the properties of the powers we
learned at the very beginning.
This is a x to the power of a, all of it to the power of b.
And according to the property which we defined at the beginning, we can write this as x to
the power of three times one over two.
So this is x to the power of three over two.
And I wouldn't take it anywhere further.
That's simple enough for me.
So it's x to the power of three halves.
Here this is a fifth root.
Fifth root of x squared.
So I have this object x squared and I'm taking the fifth root of it.
Well, fifth root in general, we can write it as well, the object is x squared and the
fifth root is one over five.
So it's the power to the one over five.
Okay.
But now I'm home and dry because we have it in the form as we can apply these laws and
properties from the beginning of the class.
So we can say this is x to the power of two times one over five, which is x to the power
of two over five.
Yeah.
Everybody following?
Yeah, some nodding there, some nodding there.
Yeah.
Cool, cool.
As I said, if you don't tell me, if you shy, wait until we finish and tell me anyway,
we need to be able to say those.
So in general, here is the formula, here is the important result of this whatever 10, 15
discussion that if I have a expression or a term in the form of end root of M, okay,
oh, there is a typo, sorry, that should be an end root of X to the power of M. Yeah.
There is a typo.
I need to change it all.
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
by N, okay.
And the other general way of writing things, if I have a square root of A and B as an object,
I can split them up and I can write it as a square root of A into square root of B. And
vice versa, both of those things works both way.
There is an equal sign in between them.
Those things on the left hand side and right hand side are identical.
Okay.
So if I look at this example 1.6, I have a square root of X divided by X to the power
of 2 times X to the power of half.
Now I can approach it more than one ways.
And this is where the uniqueness of people comes in because I may do it first, I consider
the denominator and then I consider the denominator with the numerator or I just consider all
of them at once and write them out in the terms.
So what I could do is I can just write, okay, look, square root of X, that's X to the half.
Yeah.
One over X squared, that's X to the minus 2, so that's X to the half times X to the
minus 2.
And one over X to the half is X to the minus half, so that's times X to the minus half.
Okay.
X to the minus half.
And now I use the rule that if I'm multiplying those same terms with different powers, then
I will just add up those powers.
So this is equals to X to the power of half minus 2 minus half.
This guy goes with that guy and I am left with X to the minus 2, which I can write as
X, one over X to the power of 2.
If I want to, these two things, I wouldn't say one is more simple than the other.
Yeah, so if anybody is asking you to simplify or answer as much as possible, these are equal,
at least to me.
So that's how I did it.
The other way to deal with this example will be to say, oh, I do it in two ways.
You know, I rewrite the numerator as it was, so X to the power of half, but I will first
deal with the denominator.
So I will write it as X to the power of 2 plus a half.
Yeah.
So that is then equals to X to the power of half divided by X to the power of 3 halves.
So I did like intermediate step to simplify it a little bit and now I am looking at two
terms.
So I can do what we have listed in the properties.
So this is X to the power of half plus X to the power of half minus 3 over 2.
Yeah.
And the surprise, I will get the same answer.
This is, wait, what?
Oh, that's not three halves, is it?
Why are you not shouting?
It's five.
Who said that?
Thank you.
That's the moment when you will be like, yeah, he told us to be quiet for two hours, so you
should be like waiting for these things.
And go, wow, what have you done?
How could you?
So it's a five.
No, this is still one here.
Five halves.
Yeah, it's here and there.
So five, five.
So this is X to the power of minus 4 over 2, which is X to the power of minus 2.
Same as before.
Okay.