All right, welcome to section five powers of 10. In this section we're going to be learning and talking about positive exponents of 10. negative exponents of 10. converting to powers of 10. Multiplication with powers of 10 Division of powers of 10 reciprocals with powers of 10. Addition and Subtraction with powers of 10, raising an exponent of 10 to a higher power and taking a root with powers of 10. So those are all the subjects we're going to look at. We're going to talk about them.

We're going to give you some examples. And hopefully at the end of the section, you'll have a very good grasp of What we mean by powers of 10. And again, as always, there'll be a phone number, you can contact me via email. We're there to help you to get you on to understand the subjects that we present. Alright, let's go on. Alright, in this first slide, we're talking about positive powers of 10.

And as I say, here in the first bullet, numbers greater than one can be written as powers of 10 by using positive exponents, alright, and right here, my exponents are right there. Those are my exponents. And I give you an example. And I says, Okay, let's represent 110 100 1010 Thousand 100,000 a million using powers of 10. Alright, so let's let's look at number one here, which is we're looking at number one. Well 10 to the zero power is one, in fact, any number to the zero power is one.

For instance, 20 to the zero power is one, two to the zero power is one, one to the zero power is one, nine to the zero power is one. So any number raise, or when I have I should say have an exponent right there, of zero is one. Okay, now we've got 10 to the one next, next, next one down here 10 to the One Power, my exponent is one. That exponent means 10. By itself, all right, or I can say another way I can look at this, I can say I can put a one in front of all these. So one time scan is 10 right there.

Okay? All right, let me clear the slide off here. And let's go down to two, or exponent two, which is right here. Okay, 10 squared, that's actually 10 times 10. Right? 10 times 10 is 100.

So if I in this example here, if I put one times 100, that's 100. All right, let's go to the next 110 to the third. That's 10 times 10 times 10. All right, well, 10 times 10 is 100 times 10 is 1000 right here. Okay, so my multiplier is 1000. If I put one there, one times 1000 is 1000.

All right, let's do one more 10 to the four, that's 10 times 10 is 100 times 10 is 1000 times 10 is 10,000. Right here, again, one times 10,000 is 10,000. My multiplier and my product since I'm multiplying by one is the same. That's going to change if that front number or that that number here In front is different. But the point I'm trying to make here is this. As I go down this way, what happens?

My multiplier and my product does what? It gets can times greater, right? 10 times greater. That's all I want you to know on this slide. We're going to build upon that. All right, you're not a master of positive powers of 10.

Yet, you just got a little bit of a taste of it. All right, we're gonna, we're gonna teach you it's going to become clearer. Okay. All right. We're just starting to understand this. So stay with us.

All right. So what I'd like you to do now is With that said, we want to convert powers of Convert to powers of 10. So try these and you can use the chart. Let me actually let me clear the slide off. Alright, I want you to convert to the powers of 10 year. Alright, but you can, you can use the chart here, right?

So for instance, I'm going to do the first one for you. I'm saying 100. What does that equal to in powers of 10? Well, is my product right there? What does that equal to? 10 squared right up there.

All right, I did the first one to you. All right. I'm not going to. I'm not going to give you a hand on this one here. See if you can do it. Alrighty.

Okay, I'm gonna stop the slide. And when we continue to see the answers, and I'll expand upon the bottom on the next slide. All right, here we are in the next slide. And I already did this one for you up here. Bye. This one here, 10,000 equals 10 to the fourth right there, and 100,010 to the five right there.

But if you've noticed, if you didn't use the chart, all you had to do is count the number of zeros, and that would be the number of the exponents. So for instance, 10,000, how many zeros do I have? 1234? What's my exponent for? Okay, 100,000 how many zeros 12345. What's my exponent?

Five. Done. Done. All right. All right. I didn't help you on this part here.

But let's look at this. I asked you to find 10 to the sixth. Well, if from the top how many zeros do I need? six zeros, don't I? So I put the one there and I just lay out six zeros 123456 Same thing on the bottom 10 to the fourth right here. All right, oh man, he zeros 1234 Okay, and the last 110 to the third, how many zeros 123.

So that's an easy way of figuring it out. And we we can go back and forth between the power of 10 and the whole number. The values of saying are 10 to this 10 squared by 10 to the to power is 100. Okay, so 10 to the two and 100 same value number same values, same magnitude, just a different way of presenting it, and in some cases, a different way of saying it. But the the value was absolutely the same. All right.

All right, let's go on to the next slide here. All right, now we're looking at negative powers of 10, right here. And let's look at the first bullet. numbers less than one can be written as powers of 10 by using a negative exponent. All right. All right.

Here's my again. Here's my exponent right here. All right. And the examples I'm giving you is point one 0.01 0.0 01 0.0001 0.00001 and so forth. I'll do this last one here. 0.000001.

And you can look at the last one there. All right. Now those numbers, these are decimals and decimals, the less than one, right? All right, let's clear the slide and go down here. All right, again, powers of 10. What did I tell you in the previous slide?

Any number raised to the zero power is what? One, okay. All right, look at that first slide. Again, if you don't believe me, I did it. I did it for you. Okay.

Now we're going down, we'll go in 10 to the minus One my exponent is minus one. Okay? My multiplier is point one. So again, if I put a one there one times point one is one. All right. All right.

All right now 10 to the minus two. Okay is zero.or 0.01 that's point one, times point one. That equals 0.01. Is my product or my answer product is a fancy word for answers we learned previously. Okay, so it's 0.0 110 to the minus 3.1 or zero. point one times 0.1 times 0.1.

Okay, if I put a one here, okay? One times zero dot 001, because 0.1 times 0.1 times 0.1 equals 0.001. There's my multiplier and if I've got a one, because my product or my answer, let's do one more 10 to the minus four. All right, what's 10 to the minus 4.1 times point one, times point one, times point one. What's my multiplier if I if I multiply them together, it's zero. The dot 0001 My answer is zero dot 001.

Okay, there you go. What do we know here? What have we seen so far? In the previous slide, I as my exponent increased, I got 10 times greater. Now I'm going this way, what happens? my exponents get 10 times smaller.

So and when I have a negative exponent, as I increase the value of the exponent by one I get 10 times Smaller. That's all I want you to know. Okay, so we have positive exponents that we spoke about in the previous slide. And now I just discussed negative exponents. Alright, just a different way of representing numbers. We're not changing the value of the numbers, like I said in the previous slide, just a different way of representing the numbers.

All right, as I show you here, we want to convert the powers of 10. All right, so here we go. 00 dot 001 equals minus 10. To the three, you can use the chart. All right, use the chart, but what do we know? What's a shortcut?

Well, I can put my decimal point this way, right? To the right of the one. But I know that I've got going to be smaller than one. So if I move the decimal point, three places. So the left, it's going to be a minus three. Same thing over here.

All right, 10, zero dot 01 equals 10 to the minus two. What do I have to do on this one here? Well, the first thing I've got to do is one over 1000. I've got to find the decimal equivalent to that. So I can use my calculator and what can I use? I can use the reciprocal key and we spoke about that so I can go one over 10,000 that will give my decimal equivalent and then I continue That decimal equivalent and find my negative powers of 10.

When my exponent has a negative power, okay? Over here, what we're doing is I gave you a negative exponent. Right? He had 10 to the minus 210 to the minus 410 to the minus three. And now we're just going to a decimal and again, you could use the chart. So 10 to the minus three is zero dot 001000 110 to the minus four is zero dot 0001 right here, and 10 to the minus two is zero dot 01.

All right, so in positive when we're using positive exponents, in powers of 10, we're multiplying by 10. And we're use when we're using negative exponents of powers of 10. You can think of it as dividing by 10. Okay, positive exponents, multiplying by powers of 10, or getting 10 times greater negative powers of 10. Dividing by 10, or getting 10 times smaller as my exponents increase. All right, we're good.

Let's go on to the next slide. Oh, before I go, as always, as always, if you have need some clarification, send me an email. And if you notice at the beginning and the end of each slide, there's a phone number if you want to try to give me a call. I don't pick up the block numbers. And I may not always be here, but you can leave me a message and I'll get back to you. I cannot return international calls, though.

So you may have to try a couple of times or you send me an email and I'll certainly see Certainly either schedule a time with you, or answer your answer your correspondence. All right, let's now let's go on. All right, well, in this example here, or I should say in this slide, we're going to convert numbers to powers of 10. I basically in the last, the last two slides, we were just taking on number 10, or multiples of 10 and showing you the exponent. All right, well, we know we've got a lot more numbers other than 110 100,000 10,000, and so forth, or point 1.0 1.001, and so forth. So what happens if we have a number and we want to represent that number in powers of 10.

And I give you an example here of, of 750. Well, there's many ways we can represent that. We can Say, and all of these are right, by the way, all of them are correct. There isn't a wrong way there isn't a right way with this. They're all correct. We can say 775 times 10.

And actually, even though we don't show it when I don't, when I just have my 10 there, and I don't have an exponent, it's assumed to be one. So we have 10 raised to the One Power, well, 10 to the One Power is what 1075 times 10 is 750, like I show you there. All right. We can also represent 750 by 7.5 times 10 to the two. I'll do this one here. All right.

So if I go 7.5 times 10 squared or 10 to the two, what's that equal to? 7.5? If you remember from the previous slide, what's 10 squared? It's 100. What is that equal to? 750.

All right. What am I doing? Well, how many zeros Do I have the two. Can I move that decimal point over two places, which I'm showing you right there. 750. All right.

Let's clear the slide. Let me let me expand upon the last one there. All right. 0.75 times 10 to the three Well, let's just do that one, though. Okay, well, what's 10 to the 3000? Isn't it?

Right? So 10 to the three is 1000. All right, so 0.75 times 1000. Well, what can I do? I can move my decimal point over what three places 123 put my zero there. And that's 750.

All righty. Still the same number 750. Just three different ways of representing 750. Did I change the value? No. Just three different ways of displaying or showing 750.

All right. All right, I just cleared off the slide. And we're going to look at this portion of the slide right here. All right. And what I showed you 75 over here over here, I give you a number I say, right? 1,640,000 using powers of 10.

Well, that number or the number portion of this is called the coefficient, which I show you right there. And if I go back over here, where I see the 75 that also is my coefficient. And I'll just abbreviate that right there. I don't want to spell it out. So that's my coefficient. Okay.

So the coefficient gives me I'm going to call it the base, or the the value of the number that I that I have. I mean, if I use your straight powers of 10, I can't do that I can only get 110 1000. Again. Point 1.0 1.001. So obviously, if I just look at one, one through 10, well, there's a lot of numbers in there, there's 23456789, I have to have a way of showing that and I do that through the coefficient here. So the coefficient can be a number greater than one or less than 10.

So I'm going to say C o f is greater than one or less than 10. Alright, that's my coefficient. And again, I'm going to stop here and clear this. Alright, so now I got this number. All right, 1,640,000. Well, let's make our coefficient between greater than one but less than 10.

All right. So if I do that, my coefficient than if again, if I make my coefficient greater than one, but less than 10, it's 1.64. Right here. All right. All right, but we know we've got millions 10s and millions. All right.

So to go how if you look here, my decimal point will be right there, right. How many places Do I have to the right 123456? So what's my exponent? Six, just like I show you right there. Alright. So I, I take my number, I find my coefficient, and I just told you coefficient wants to be greater than one, but less than 10.

I've got 1.64 1.64 is greater than one and certainly less than 10. So there's my coefficient. And then I got to have to figure out how many decimal places or how, how big that is, how big that number is in powers of 10. All right, and just look at that. Here, before we go on, when no coefficient is given, it is assumed to be one. All right in powers of 10, you're going to hear about two different type of notations, scientific notation, and engineering notation.

All right. Again, they do not change the value of the number. It's how the number is represented using or in terms of powers of 10. So when I'm using scientific notation, I want the coefficient of one or more, but less than 10. So for instance, an example a scientific notation would be three dot three times 10 to the four, my coefficient is right here. greater than one, but less than 10.

Then I adjust my powers of 10. Engineering notation, we want to have powers and 10 have multiples of 336 912 minus three minus six minus nine minus 12. And if you take some electronic courses with me, you'll see why that is I'm not going to get into it here. But just quickly minus three is the represents Milly minus nine, I'm sorry, minus six represents micro. So we have milli volts, micro volts milliamp ers micro amperes. And that's all I'm going to say here.

But that's why in engineering, we want to use have three and when you take the courses that will be a little bit clearer. All right. But again, scientific notation coefficient wants to be greater than one but less than 10 engineering notation, I want to have my powers of 10 in multiples of three. And then in engineering notation, I will adjust my coefficient to get the correct value of the number. All right. So why don't we do this real quick.

Let's take 15,000 and write that both into engineering terms. And also into I'm sorry, write that in, in scientific notation, and also in engineering notation. All right. All right. You'll see the answers on the next slide and I'll go over all right, so now I asked you to take 15,000 and write it both into it. Engineering terms, which is scientific and engineering well, is my 15,000 right there.

What do we know, scientific notation? I want to be greater than one but less than 10. Well, 1.5 that's 15,000. What do I have to multiply 1.5 by to get 15,000 10,000. What's 10,000? Go back look at the chart.

It's 10 to the four. So it'd be 1.5 times 10 to the four. Okay. The bottom one here, next one engineering notation. What did we say we want multiples of three. We've got 15,000.

If you look at the beginning charts that I gave you, 1000 is white. 10 to the 315 times 10 to the three because 10 to the three equals 1000 15,000. Right there. All right. All right. We've got some examples in this in these slot, not example, some exercises in these slides, some homework, or some additional work the week.

I'm old fashioned, I call it homework. I'll just say, additional exercises. And again, you know how to get a hold of me. All right, this is going to end this chapter in powers of 10. We've got another one coming up. The next lecture is going to expand upon this.

It's going to be addition, subtraction, multiplication, division of powers of 10. All right, so we'll see in the next lecture, take care. Talk to you soon.