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Video transcript

let's talk about multiplying by ten a hundred and a thousand there's some cool number patterns that happen with each of these so let's start here with something like four times ten one that maybe we're comfortable with or already know four times ten would be the same as saying for tens for tens and for tens one way we could represent that is a ten plus a second ten it's a third 10 plus a fourth 10 or four tens and now let's count that 10 plus 10 is 20 plus 10 is 30 plus 10 is 40 so our solution is 40 or a four with a zero and this is the pattern that we've seen before when we multiply 4 times 10 we keep our whole number of 4 and we add a zero to the end for the times 10 so another example of that might be something like 8 times 10 well 8 times 10 is the same as 8 tens and this time let's just count them if we count eight tens it'll be 10 20 30 40 50 60 70 80 so when I counted eight tens the solution was 80 or an eight with a zero on the end so times 10 when we multiply a whole number times 10 the pattern is that we end up adding a zero to the end of our whole number so let's take now what we already know about tens and let's apply it to hundreds something like let's say 2 times 100 there's a couple ways we can think about this one way is to say that this is the same as two hundreds two hundreds which is 100 plus another hundred there's quite literally two hundreds which is a total of two hundred or two with two zeros on the end now we have two zeros on the end or another way to think about this is to two times 100 instead of saying times 100 we could say times 10 times 10 because 10 times 10 is the same as a hundred and two times ten we know is a two with a zero on the end which is 20 and 20 times 10 then will be 20 with a zero on the end because we multiplied by 10 twice we added two zeros and multiplied by 100 is just that it's exactly that it's multiplying by 10 twice so if x 10 adds 1 0 then times 100 or times 10 twice adds two zeros to our answer and we can go even further and think about thousands let's try something like 9 times 1,000 well we could think of this as 9 thousands and if we have nine thousands then we have 1,000 2,000 3,000 4,000 5,000 that was 5 6,000 7,000 8,000 9,000 so when I counted a thousand nine times our solution was nine thousand or looking at the numbers a nine our original whole number with three zeros after it so nine times a thousand is nine thousand or nine with three zeros and we can go back to what we did before of thinking about this in terms of tens we've worked out why multiplying by 10 adds a zero so let's think about 1,000 in terms of tens 1000 is equal to 10 times 10 times 10 10 times ten is a hundred and 110 is a thousand so instead of 1000 we can write 10 times 10 times 10 these are equivalent and so when we multiply a number times 10 we add a zero but here we're multiplying by three tens so we add three zeros so let's look at that all as one pattern let's say something let's take seven and number seven and let's multiply it by ten by a hundred and by a thousand and see what happens seven times ten is going to be seven with one zero because we have 110 seven times a hundred will be seven with two zeros because again 100 is the same as ten times ten so this is seven times ten twice so we have two zeros and seven times a thousand will be seven thousand or seven with three zeros because a thousand is equal to 10 times 10 times 10 or three tens so we add one two three zeros and so we can see the pattern here when we multiply by ten which has one zero we add one zero to the end of our whole number when we multiply a whole number times a hundred which has two zeros we add two zeros for hundreds and for thousands when we multiply by one thousand we'll add three zeros to the end of a whole number

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Multiplying 1-digit numbers by 10, 100, and 1000 (video) | Khan Academy

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If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

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- [Instructor] In this video, I'm going to introduce you to a new type of mathematical notation that will seem fancy at first, but hopefully you'll appreciate is pretty useful and also pretty straightforward. So let's just start with some things that we already know. So I could have just a 10. I could take two 10s and multiply them together, so 10 times 10, which you know is equal to 100. I could take three 10s and multiply them together. 10 times 10 times 10 is equal to 1000. And I could do this with any number of 10s. But at some point, if I'm doing this with enough 10s, it gets pretty hard to write. So for example, let's say I were to do this with 10 10s. So if I were to go 10 times 10 times 10 times 10, that's four, that's five, that's six, that's seven, that's eight, that's nine, that is 10 10s. Let's see, one, two, three, four, five, six, seven, eight, nine, 10. This is going to be equal to, even the number that it's equal to is going to be quite hard to write. It's going to be one followed by 10 zeroes. One, two, three, four, five, six, seven, eight, nine, 10. We put the commas there so it's just a little bit easier to read. This right over here is 10 billion, and it's already getting kinda hard to write, and imagine if we have 30 10s that we were multiplying together. So mathematicians have come up with a notation and some ideas to be able to write things like this a little bit more elegantly. So the way they do this is through something known as exponents. Exponents. And so 10 times 10, we can rewrite as being equal to, if I have two 10s and I'm multiplying them together, I could write this as 10 to the second power. That's how someone would say it. They would say 10 to the second power. That looks fancy, but all that means is let's take two 10s and multiply them together and we're going to get 100. Just so you're familiar with some of the parts of this, the two would be called the exponent and the 10 would be the base. So 10 to the second power is 10 times 10 is equal to 100. So how would you write 10 times 10 times 10 or 1000? How would you write that using exponents? Pause this video and see if you can figure that out. Well, as exactly as you might have imagined, we're taking a certain number of 10s and we see we're taking three 10s and we're multiplying them together. So this would be 10 to the third power. 10 is the base, three is the exponent. We would read this as 10 to the third power. If you ever saw 10 to the third power, that means hey, let me multiply 10 times 10 times 10. That's the same thing as 1000. So this is really another way of writing 1000. And what about this number here, 10 billion? What's a way that we could write it using exponents? Pause the video and figure that out. Well as you might have imagined, we were taking 10 10s and multiplying them together. This is 10 to the 10th power. And we can go the other way around. If someone were to walk up to you on the street and say what is 10 to the fifth power? What is that? What number that you're probably familiar with would this be? Well this would mean that we're going to take five 10s and multiply them together. So 10 times 10 times 10 times 10 times 10. And so 10 times 10 is 100, 100 times 10 is 1000, 1000 times 10 is 10000, 10000 times 10 is equal to 100000. So there you have it. You have the basics of exponents when we're dealing with 10, and I know what you were thinking. Can I put another number here instead of 10? And the simple answer is, you can, but we're not gonna cover that just yet in this video.

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Introduction to powers of 10 (video) | Khan Academy

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Instructor] In this video, I'm going to introduce you to a new type of mathematical notation that will seem fancy at first, but hopefully you'll appreciate is pretty useful and also pretty straightforward. So let's just start with some things that we already know. So I could have just a 10.

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Multiplying 2-digits by 1-digit with partial products (video) | Khan Academy

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We can multiply larger numbers by multiplying by the tens and ones separately, then adding the products together. Multiplying with partial products is one way we use the distributive property.

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Scales of the Universe in Powers of Ten - Full HD 1080p

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Multiplying 2-digit numbers (video) | Khan Academy

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Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

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We now have the general tools to really tackle any multiplication problems. So in this video I'm just going to do a ton of examples. So let's start off with-- and I'll start in yellow. Let's start off with 32 times 18. Say 8 times 2 is 16. Well, I'll do it in our head this time because you always don't have all this space to work with. So 8 times 2 is 16. Put the 1 up there. 8 times 3 is 24. 24 plus 1 is 25. So 8 times 32 was 256. Now we're going to have to multiply this 1, which is really a 10, times 32. I'll underline it with the orange. 1 times 2-- oh, we have to be very careful here. 1 times 2 is 2. So you might say hey, let me stick a 2 down there. Remember, this isn't a 1. This is a 10, so we have to stick a 0 there to remember that. So 10 times 2 is 20. Or you say 1 times 2 is 2, but you're putting it in the 2's place, so you still get 20. So 10 times 2 is 20. It works out. Then 1 times 3. And we have to be very careful. Let's get rid of what we had from before. 1 times 3 is 3. There's nothing to add here, so you just get a 3. And so you get 10 times 32 is 320. This 1 right here, that's a 10. 10 plus 8 is 18. So now we just add up the two numbers. You add them up. 6 plus 0 is 6. 5 plus 2 is 7. 2 plus 3 is 5. Let's keep going. Let's do 99 times 88. So a big number. 8 times 9 is 72. Stick the 7 up there. And then you have 8 times 9 again. 8 times 9 is 72, but now you have the 7 up here. So 72 plus 7 is 79. Fair enough. Now we're done with this. Let's just delete it just so that we don't get confused in our next step. In our next we're going to multiply this 8 now times 99. But this 8 is an 80. So let's stick a 0 down there. 8 times 9 is 72. Stick a 7 up there. Then 8 times 9 is 72. Plus 7 is 79. 2 plus 0 is 2. Let me switch colors. 9 plus 2 is 11. Carry the 1. 1 plus 7 is 8. 8 plus 9 is 17. Carry the 1. 1 plus 7 is 8. 8,712. Let's keep going. Can't do enough of these. All right, 53 times 78. I think you're getting the hang of it now. Let's multiply 8 times 53 first. So 8 times 3 is 24. Stick the 2 up there. 8 times 5 is 40. 40 plus 2 is 42. Now we're going to have to deal with that 7 right there, which is really a 70. So we got to remember to put the 0 there. 7 times 3, and let's get rid of this. Don't want to get confused. 7 times 3 is 21. Put the 1 there and put the 2 up here. 7 times 5 is 35. Plus 2 is 37. Now we're ready to add. 4 plus 0 is 4. 2 plus 1 is 3. 4 plus 7 is 11. Carry the 1. 1 plus 3 is 4. 4,134. Let's up the stakes a little bit. So let's say I had 796 times 58. Let's mix it up well. All right, so first we're just going to multiply 8 times 796. And notice, I've thrown in an extra digit up here. So 8 times 6 is 48. Put the 4 up there. 8 times 9 is 72. Plus 4 is 76. And then 8 times 7 is 56. 56 plus 7 is 63. I'm sure I'll make a careless mistake at some point in this video. And the goal for you is to identify if and when I do. All right, now we're ready, so we can get rid of these guys up here. Now we can multiply this 5, which is in the 10's place. It's really a 50. Times this up here. Because it's a 50 we stick a 0 down there. 5 times 6 is 30. Put the 0 there, put the 3 up there. 5 times 9 is 45. Plus the 3 is 48. 5 times 7 is 35. Plus 4 is 39. Now we're ready to add. 8 plus 0 is 8. 6 plus 0 is 6. 3 plus 8 is 11. 1 plus 6 is 7. 7 plus 9 is 16. And then 1 plus 3 is 4. So 796 times 58 is 46,168. And that sounds about right because 796-- it's almost 800. You know, which is almost 1,000. So if we multiplied 1,000 times 58 we'd get 58,000. But we're multiplying something a little bit smaller than 1,000 times 58, so we're getting something a little bit smaller than 58,000. So the number is in the correct ballpark. Now let's do one more here where I'm really going to step up the stakes. Let's do 523 times-- I'm going to do a three-digit number now. Times 798. That's a big three-digit number. But it's the same exact process. And once you kind of see the pattern you say, hey, this'll apply to any number of digits times any number of digits. It'll just start taking you a long time and your chances of making a careless mistake are going to go up, but it's the same idea. So we start with 8 times 523. 8 times 3 is 24. Stick the 2 up there. Now 8 times 2 is 16. 16 plus 2 is 18. Put the 1 up there. 8 times 5 is 40. Plus 1 is 41. So 8 times 523 is 4,184. We're not done. We have to multiply times the 90 and by the 700. So let's do the 90 right there. So it's a 90, so we'll stick a 0 there. It's not a 9. And let's get rid of these guys right there. 9 times 3 is 27. 9 times 2 is 18. 18 plus 2 is 20. And then we have 9 times 5 is 45. 45 plus 2 is 47. I don't want to write that thick. 47. Let me make sure I did that one right, and let's just review it a little bit. 9 times 3 was 27. We wrote the 7 down here and put the 2 up there. 9 times 2 is 18. We added 2 to that, so we wrote 20. Wrote the 0 down there and the 2 up there. The 9 times 5 was 45. Plus 2 is 47. You really have to make sure you don't make careless mistakes with these. Then finally, we have to multiply the 7, which is really a 700 times 523. When it was just an 8 we just started multiplying here. When it was a 90, when we were dealing with the 10's place, we put a 0 there. Now that we're dealing with something that's in the 100's, we're going to put two 0's there. And so you have 7-- and let's get rid of this stuff. That'll just mess us up. 7 times 3 is 21. Put the 1 there. Stick the 2 up there. 7 times 2 is 14. 14 plus our 2 is 16. Put the 1 up there. 7 times 5 is 35. Plus 1 is 36. And now we're ready to add. And hopefully we didn't make any careless mistakes. So 4 plus 0 plus 0. That's easy. That's 4. 8 plus 7 plus 0. That's 15. Carry the 1. 1 plus 1 plus 1 is 3. 4 plus 7 plus 6. That's what's? 4 plus 6 is 10. It's 17. And then we have 1 plus 4 is 5. 5 plus 6 is 11. Carry the 1. 1 plus 3 is 4. So 523 times 798 is 417,354. Now we can even check to make sure. And so this is the moment of truth. Let's see if we have-- let's see. 523 times 798. There you go. Moment of truth. I don't have to re-record this video. It's 417,354. But we did it without the calculator, which is the important point.

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Multiplying multi-digit numbers (video) | Khan Academy

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Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.