All right here we're going to talk about raising an exponent of 10 to a higher power. So I give you an example here, two times 10 squared. And we're going to cube that. All right, so we break out the coefficient, which, if you remember is the two. All right, and we're going to cube that because it's inside the bracket. So here we go here, we know that two cubed is two times two times two, which is eight.

All right, and then all we do for the powers of 10 is multiply the exponents in this case, two times three or three times two. And we get a Six. So in powers of 10, we have two cubed times 10 to the sixth power. All right, and we again we multiply two times two times two, we get eight times 10 to the sixth. And then when he when we expand upon that, we get 8 million because eight times 10 to the sixth is 8 million. That's it.

That is it. And I've given you some exercises here. So please hit the pause button, do the exercises. And as always, on the next slide, you'll see the answer. Okay, here are the answers here on this one, it's one times 10 to the sixth and down this one here is one times 10 to the minus Sixth, nobody here we got four times 10 to the fourth. I'm just gonna take a minute here and expand upon this one.

So if you notice we've got two times 10 squared in brackets that are all squared. So what do we have here? Well, that's actually two squared, right? times 10. And what do I do to the exponents? I multiply these two exponents right there.

So two times two is four. All right. So now going out two squared is four times 10 to the fourth. All right. And that's it. If I want to expand what's 10 to the fourth?

It's 10,000. So this is answer if I want to bring it out, is 40,000. Right there. All right. There's my answer. We've done it.

We're ending this We're going on to the next one, seeing the next next slide. All right here on this slide, we're going to talk about taking a root with powers of 10. Right there. And the rules are, take the root of the coefficient, and then divide the exponent by the indicated root. So I give you an example here. What I asked you to do, or what we're going to do together here is find the root of eight times 10 to the sixth.

Okay, the third root, the cube root. So here we are here. So what we need to do is we break out the coefficient and the powers of 10. So we're going to find the cube root of eight. And then we're going to find the cube root of 10 to the six. All right, so let's do the coefficient.

All right, the cube root of eight is two And then all I do when I find the cube root of this coefficient is i divided by the roots. So in this case, the power is six to the power of 10 is six, I'm looking to find the cube root of three. So I just divide six by three. And my answer is two times 10 squared. Okay, because six divided by three is two, as my answer to right there. And so if I want to expand this into a whole number, it would be what?

200? That's my answer. All right. Now let's look at another one here in the bottom. And what happens if the power of 10 is not an even number? Well, what we need to do is we need to adjust that that power So the example here is font, let's find the square root right here, of 40 times 10 to the five, so we need to adjust the power of 10 to an even number.

So we could either go up to six down to four, so we were going to go up to six. All right, so where is my decimal? When I go here, my decimal is right there, even though we don't show it, it's assumed to be there. So what am I doing? moving my did to go I'm moving my decimal point to the left to adjust my coefficient. Alright, so I can get an even power of 10.

So like I tell you here, when I move my decimal point to the left, which I've done here, I increase the powers of 10 by one. So you'll notice right here, I've got 10 to the fifth power, I've moved my decimal point to the left, which makes my coefficient four. And now I have to adjust my power of 10. In this example, by one because when I move my decimal point to the left, I increase the powers of 10 by one, okay? If just just to follow this throw here, if I move my decimal point to the right, meaning I go this way, then I decrease my powers of 10. So that's a benchmark.

When I move my decimal point to the left, I increased by Have 10 by one. And when I move my decimal point to the right, I decreased powers of 10. And let me clarify one more thing. Okay, every time I move my decimal point, one place to the right or one place to the left, I either increase by one, or decrease by one. When I move my decimal point, two places to the right or two places to the left, I increased by one and I increased by two, or decreased by two. So how I decrement are increased my powers of 10 is dependent upon how many places I move my decimal point, I move my decimal point by two I either increased by two or decreased by two depending on which way I move my decimal point.

I move my decimal point three places, either to the right or to the left. I either increase my powers of 10 by three or decrease my powers of 10 by three, depending on which way I go. Alright, so make sure you understand that it will make it very easy for you. When you look at these in an application and you have to move your decimal point either either to the right or to the left, you'll know which way to go. All right, so let's finish this. So over here, I've got the square root of four times 10 to the sixth.

So now I just break it up, what's the square root of four is two. And now all I do is divide my exponents because I adjusted my powers of 10 to an even number. So six divided by two is three. So my answer is two times 10 to the third and if I want to expand that into a whole number, it would be 2000 i. Okay, let's stop here and go on to the next slide. All right, let's take a few minutes, hit the pause button and let's do these three exercises here.

Okay, square root first one is square root of four times 10 to the eighth. The second one, find the cube root of 27 times 10 to the ninth, and the last one 3.6 times 10 to the seventh to the half power, isn't the half power the square root? And in this example, don't we have to use our policies or our rules on changing the power of 10? All right, take a look at them and as always, the answers will be on the next slide. All right, here are the answers. Two times 10 to the Fourth, three times 10 to the third.

And over here we got six times 10 to the third. Okay, we had to adjust our decimal point. So we had to move the decimal point on 3.6 this way, and that became 36. And when I move my decimal point one place to the right, I decrease my powers of 10 by one, so that came to be times 10 to the sixth. The square root of 36 is six times what? six divided by we got 10.

And we want to divide by exponents. So what do we have? We've got 36 times 10 to the third because six divided by two is three. That's it. Okay. So let's go on to the next slide.

And as you know, there's a phone number at the beginning of this slide presentation. And at the end, you can also send me an email through this platform. I'm here to help. And I've talked about how I did that before, but I don't respond to block numbers. I don't return international calls. But if you are, need some help from me, and you're International, and we'll work it out, we'll set up a time we'll do something.

All right. Okay, take care. See in the next slide. Okay, before I go on, I want to just want to correct that right there. That should be Six, not 36 that's the correct answer. So, sometimes when I am explaining things, I'm concentrating more on my explanation then somehow, maybe the proper answer, but again, I made a mistake, it should be six, not 36.

Alright, see you in the next slide. Okay, this ends this section on powers of 10. The next section we're going to talk about is logarithms. So we'll see over there, take care