Alright, powers of negative numbers. All right, as we stated before, raising a number to a power is the same as repeated multiplication right here. All right? All right. So even number of negative signs, the answer is positive. And an odd number of negative signs, the answer is negative.

And I show you some examples here, for instance, minus three squared, well minus three times a minus three. Well two is what is positive, so my answer is a positive nine. All right, you can pull out the calculator and do it. Just pull out three times minus three times a minus two is nine. Actually we'll stop here. I'll do this one for you.

So let me stop this recording and get my count. All right, I've got my calculator down. And all I'm going to do here it's in standard mode. It's the one I use with the operating system or the one that comes with the operating system. So I'm going to hit clicking three, and this means change sign. So I'm going to change the sign to minus three.

And I'm going to hit the square key plus nine. All right, that's it. All right, let's clear it. Let's do this one. Okay, since I've got it down, let's do this one, minus two to the four, again, even number of minus signs, so my answer should be positive. So let's go up here.

We're going to have to change that. Let's go to scientific mode. And we're going to use this this function here. So again, I'm going to plug in to change signs to negative. And I, now I want my power. That means I want my power.

So I just put in the minus two here, right there. So now it wants to power and right here we want four. So I put in the four and now I'm going to say equal, and it's a plus 16. Just as I show you right here. All right. Okay.

All right. Now let's move that and let's, well let's move this and go over here. Okay, I've repositioned my calculator, and I'm still in scientific mode. Alright, so let's do this example minus two to the third power. All right, so again, plugin groups. Let's, let's clear this.

Okay, so now I play getting too. All right. I'm using this function here. Ah, wait a minute. Let's, let's start again here. Let me clear it.

So I'm going to put into change the sign. All right, hit that. And now I'm going to put in my power. In this case, it's three. Three equals minus eight. Now there's my answer right there, minus eight.

So we know that that's working good, are working correctly, I should say. So let's clear this again. All right, let's do minus three raise to the five power again. I'm putting in three, change my sine x to the Y, I just found the x function. Now I'm looking for the Y or the power five, enter minus 243 right? They're right there.

So it's it's pretty straightforward using the calculator. And again, just remember these rules. If I have an even number of minus signs, my answer is positive. If I have an odd number of minus signs, the answer is negative. All right? What I'd like you to do is stop the recording I've given you four quick examples scare, pull out your calculator and and just see if you get them and then when you continue with this presentation, the answers will be on the next slide.

Okay, here are the answers right here and if you have not gotten them our look at the previous As explanation again, and then you can always give me a call or send me an email. We'll try to help you out. But this is pretty, pretty straightforward. Alright, so let's go on to the next topic now. Okay, well, we're talking about roots of negative numbers now. And as we said before, since an even power of negative numbers always lead to a positive number answer.

And an odd power of negative numbers always lead to a negative number answer. We got to look at this. So for instance, when I want to find the cube root of eight, or minus eight, I should say I. It's got to be a negative number. In other words, the answer is got to be negative. Because if I have an even number, what happens?

The answer will always be positive. So when I find the cube root, okay, that will mean how many times a number multiplies by itself will give me the answer in the center of the radical sign in this place a minus eight. All right. So, it's going to be a minus two because minus two times a minus two is a plus four times a minus two will equal a minus eight. All right. So when we find the The roots of negative numbers.

It has to be odd, 3579 and so forth. All right. I give you an example of another one right here. All right, and I'm looking for a number. That will be multiplied by itself five times. That will give me the answer of A minus 3125.

All right, that happens to be a minus five and I, I show you down here that that is true, because if I multiply minus five, five times I get the answer here. So let's clear the slide here for a minute. Okay, well, I've brought down my, my route calculator and gave you a link right here for it. So if just to prove something out here. I'm going to say that we want a cube root. And again, we've gone through this before.

And I'm going to plug in minus eight. So let's calculate that. Ah minus two. All right. All right. So we know that that's working now.

All right, let's reset this declare it. And let's look at the other one that I did. And again, five, okay, minus 3125. And I want to know what that is. So minus five minus five. But let's just for the heck of it, and we're going to look at this in the next slide.

What happens if I want to find let's, let's do that, let's actually let's do this. Let's go with eight. Okay? And let's plug in minus eight. And let's say what the square root. And let's calculate that.

Let's see what happens. Doesn't work. Because I don't have a number multiplied by itself. That will give me a minus eight to positive numbers will give me a plus answer. And two negative numbers will give me a plus answer. All right, let's just do one One more.

And then we're going to review this again on the on the next slide. But let's go 16. All right, and let's change that to four. And let's count out, let's, let's, let's clear that. Let's do four minus 16. Again, and let's calculate it again.

Right? Because for, I'm looking for what number multiplied by itself will give me a minus 16. Right there. It's not going to happen because again, going back to our rules, since an even power of negative numbers always lead to a positive answer. All right. Okay, so we've we've seen that and here is the here right there is the URL for that calculation.

And take a look at it if my link doesn't work, I use it a lot. And it's very good. There's there's other ones on there, you might like them better. I happen to like this one. Okay, thank you so much. And we'll see on the next slide.

All right, and this slide here, we're going to talk about even routes for negative numbers. And if you remember from the previous slide, we can't do that. Because if we have a positive number of negative signs, the answer is always positive. So as I say, here, in the first first bullet, we cannot calculate the square root of a negative number. We assume that the square root numbers do exist for negative numbers, so we assume that all right, these roots are called imaginary numbers and the magic unary numbers are on a j access, okay, which are a vertical access on the number line, which we're not going to show because I don't want to get that complicated. It really is, if you're going to get very, very deep into scientific applications, which this math course is not about, you would need to know that.

So all we need to know right now for this particular course is that when I'm trying to find a root, an even number of roots for a negative number, they called an imaginary number. And they're represented on the J x s. All right, and that j represents the square root of A minus one. All right. So basically, I'm giving you an example here. We want to find out The square root of minus four, well, we kind of have to break it up a little bit. And you don't really have to do this all the time, I'm going to show you a, a quick way of doing it, you assume something, but just to go through the explanation.

Okay, when I have that minus four, that will equal this a minus square root of A minus one times the square root of four. Okay, we tell you up here that j represents a minus one. So now we have this here. All right, and the square root of four is two. So what we're saying here is the square root of A minus four is a J, two. All right, and j represents an imaginary number.

That's it, and we still can use our calculator but we just need to assume that it's an imaginary number. So for instance, and let lets us, let me get my calculator down. Alright, I've got my calculator down, I mean using the calculator that comes with the operating system in standard mode. And here's my square root. So if I want to find the square root of minus four, and let's up, let's just do it, let's say, for change signs, I'm going to hit the square root, invalid input. All right, we know that it's not going to work.

So let's clear it. But I know that it's a minus four, and J equals A minus one. So all I do is say four square root and that equals two. I have to remember that that's an imaginary number. And I need to put the J in front of it. And I also need to remember that the J represents a square root of A minus one.

That's it. That's it.