So the first algebra we will consider is a indices or powers.
So indices provide a convenient way to write a expressions or terms
when you multiply a same quantity or same number or same variable multiple times.
So if you say oh three times three, cool, that's kind of easy going.
But if you will think of oh I want to multiply three twenty times,
then it's like oh three times three times three times three and I'm already lost how far did I go.
So I have to count again and it is another 15 of them to go.
It's very tedious.
So and mathematicians in general are very lazy people.
So what we do is we're making our life easy.
So instead of writing three multiplied with each other for twenty times as a three times three times three times all those things,
we find something shorter to do that and we write it as a three to the power of twenty.
Yes, we do the power of twenty.
So couple of example how this works three to the power of two, eight to the power of two, B to the power of three, everybody following this.
Yeah, I promise a little bit easy.
In general, I take a natural number and as a power, then extra the power of n will be x times x times x multiplied n times.
Okay, n times.
So that's a natural number.
A couple of examples you welcome to read through those be careful.
What do you taking power off?
Okay, if I will take extra the power of three.
If I will say extra the power of three, that is sorry x like this, I take x and take the power of three, that's extra the power of three.
So that's fine. Yeah, if I will take three of x to the power of three, that's a three x to the power of three, that's still fine.
Everybody okay with that.
Now the problem I'm having here is that I'm taking three x that whole thing to the power of four in that example.
So I have to multiply that whole thing with itself four times.
Okay.
And then you need to keep in mind over three to the power of four.
I have to do that as well.
So you will have 81 times extra the power of four.
And that's where the usefulness of the brackets comes in.
Okay, so if I want to indicate that I want to take a power of this whole thing, and it is more than just a variable or more than just a number.
I stick the bracket in there and make it to create everybody that I mean all this whole thing.
Okay, so these are the powers and here is a bunch of properties associated with those which may be useful for you to remember.
Remember, if you know them, good.
If you don't know them or you finding yourself making mistakes when applying them, then stick it on a little piece of paper, you know, property indices, write them down, stick it on the fridge and whenever you go and get a little bit of a cheese, you will see it there.
And it will help you to memorize them.
Eventually you will throw it off because you have been using it for a while.
But you want some reminder.
Okay, so a couple of properties, especially these when we start to deal with a inverse powers.
Okay, inverse is the negative power.
So extra the power of my A in there means it's one over extra minus a.
Okay, one over extra minus a.
So these, there is an equal sign in between those, you know, so these are equations.
These things because they are equal, they are identical, they mean the same thing.
These very straightforward.
So couple of examples to the power of three times to the power of two is to the power of and if I look at the property it says here.
Yeah, it's a plus B.
So it's the first part of plus the other power.
So extra the power of three times extra the power of two will be sorry to the power of three times to the power of two will be to the power of three plus two.
So that is to the power of five.
Okay.
Good.
A couple of more examples here. This is a different one. So this is three to the power of two to the power of four. Okay.
So if I don't know, I look at the property.
And it will tell me it's this second case extra a whole of it to the B.
So this is then according to the rules three to the power of two times four, which is equals three to the power of eight.
Okay.
Two times four.
Good.
Three to the power of minus three. That is nothing to really do that is one over three to the power of three.
You may go as far as this is one over 27.
If you want.
Okay.
Everybody is following this.
Can you all see this?
And it's another thing.
Any technical issues? I mean, if you tell me you can't see it, I will tell you to come here, you know, come close.
Can you hear me? Good guys in the end.
Good. I am a little bit higher with the voice. I wonder what that guy is going to do when I will be watching this up.
Okay.
So there we go.
A couple of more. I will skip them for you to do. Okay.
Same principles, same properties.
I will consider.
I will consider this one five to the power of 11 divided by five to the power of nine. Okay.
So I can write this according to those properties as five to the power of 11 times five to the power of minus nine.
Because it's in the denominator of that fraction.
And then I can use the other property.
I have to add those things together if I want to combine it.
So this is five to the power of 11.
Now, plus the other one, but careful, there is a negative quantity.
Yes, I have to keep that in mind.
So this is 11 minus nine.
Yes, that's where the minus is coming from.
And that is then five to the power of two.
Okay, you can take a calculator and go 25 if you want to.
Good couple of more.
If you will not be able to keep up writing, then that's the moment you shouted me.
Yeah, moment or one second.
A couple of more examples here.
These are now equations. Yeah, this is actually quite cool.
So there is an equal sign here, for example, saying that the left hand side is equals to the right hand side.
Okay, now there is a unknown unknown X and we want to solve it for X.
So we are getting to this mathematical object X unknown.
We will refer to them as a variables as soon as we get to the functions, but at this moment, we just talk about it as unknown, you know, so we want to find out what that is.
Okay, what is this unknown.
We use the knowledge we just developed or recap about the indices, and we look at the left hand side.
Yeah, left hand side of this equation that is a three to the power of 12 divided by three to the power of eight.
That is according to the properties three to the power of 12 times three to the power of minus eight because three to the eight in denominator is three to the minus eight.
The next property will tell you that if you have a dose two with the same base three, you can add up these indices, these powers.
So we can write it as a three to the power of 12 minus eight, which is equals to three to the power of four.
Yeah, and if I will just remind myself that the right hand side is equals to three to the x the only way how that will happen is if x is equals to four.
Okay, so the unknown hands.
X is equals to four.
Okay, X is equals to four.
There may be a problem like that to deal with.
Okay.
If you find it easy, it's good.
But bear with me.
You have to go through those things.
There are a couple of answers.
I do produce like answers if I have the problems there, but I do not show the working in in the gap versions.
Okay, so that's, that's the indices in in the form of a whole number so far.