# data[].attributes.headline
Multiplying and Dividing Whole Numbers by Powers of 10

# data[].attributes.subject
Multiplying and Dividing Whole Numbers by Powers of 10

# data[].attributes.attachment_link.description
When you visit any website, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized web experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and change our default settings. However, blocking some types of cookies may impact your experience of the site and the services we are able to offer. More information

# data[].attributes.headline
Multiplying and Dividing Decimals by Powers of 10

# data[].attributes.subject
Multiplying and Dividing Decimals by Powers of 10

# data[].attributes.attachment_link.title
Oops. Something went wrong. Please try again. | Khan Academy

# data[].attributes.attachment_link.description
Uh oh, it looks like we ran into an error. You need to refresh. If this problem persists, tell us.

# data[].attributes.attachment_link.readable_text
We've already seen that a fraction like 2/9 can be interpreted as 2 divided by 9. Let me do the 2 in the same color. That if we have a fraction, it could be interpreted as the numerator divided by the denominator. And this leads to all sorts of interesting conclusions, some of which we've already seen, and some of which are a little bit new. So, for example, if I had the fraction 7/7, this can be interpreted now as 7 divided by 7, as our numerator divided by our denominator. And 7 divided by 7 is of course equal to 1. And this is consistent with what we've already seen. 7/7 would get us to a whole, and a whole is the exact same thing as one. But we could do things a little bit more interesting as well. We could take something like 18/6 and realize, wow. This is the same thing as 18 divided by 6, which we know is equal to 3. And we should do a little reality check. Does this make sense, that 18/6 should be equal to 3? Well, we could rewrite it. We could rewrite 18/6. Let me make the sixths that same orange color. That's going to be the same thing. 18 is 6 plus 6 plus 6. And then all of that over 6, and then that's the same thing. That's the same thing as 6/6 plus 6/6 plus 6/6. And I could make this right over here in orange. And we've already seen, or we've seen many, many videos ago, that 6/6, just like 7/7, these are each equal to a whole. These are each equal to 1. And we can now view this as 6 divided by 6, which is the same thing as 1. So this is 1 plus 1 plus 1, which is, of course, equal to 3. But this starts to raise an interesting question. This all worked out just fine because 18 is a multiple of 6. 6 divides evenly into 18. But what happens if we start having fractions where the denominator does not divide evenly into the numerator? Let's say we have a fraction like 23 over 6. Well, we know that we can interpret this as 23 divided by 6. And if we actually divide 23 by 6-- let's do that. So we divide 6 into 23. We know 6 goes into 23 three times. 3 times 6 is 18. And then when you subtract, you end up with a remainder of 5. So we might say, hey, 23 divided by 6 is equal to 5 remainder 3. But that's not that satisfying. What do I do with this remainder? This really isn't a number here. This is just saying that we're going five times, and then we have a little bit left over. What we can do now is manipulate this a little bit so that we can realize that this is a number, and in particular, a mixed number. So for example, we could start with a 23 over 6, and we could divide it into-- or we could decompose the numerator into one part that is divisible by 6, evenly divisible by 6, and the remainder. So for example, 23 over 6 we can rewrite as 18 plus 5 over 6. Notice I decomposed the 23 into one part that is a multiple of 6, and it's the largest multiple that essentially fits into 23, or that is less than or equal to 23, and then the remainder. When you divide 6 into 23, you get a remainder of 5. You could view it as, I divided it into the remainder and everything else. And the reason why this is interesting is because we know that this is going to be equal to 18 over 6 plus 5 over 6. Well, we already know that 18/6 is the same thing as 18 divided by 6, or 3. So this is the same thing as 3. So we know that 23 over 6, which is the same thing as 18 plus 5 over 6, is the same thing as 3 plus 5/6. Or, if we want to write it as a mixed number, we could write it as 3 and 5/6.

# data[].attributes.attachment_link.title
Creating mixed numbers with fraction division (video) | Khan Academy

# data[].attributes.attachment_link.description
We've already seen that a fraction like 2/9 can be interpreted as 2 divided by 9. Let me do the 2 in the same color. That if we have a fraction, it could be interpreted as the numerator divided by the denominator.

# data[].attributes.attachment_link.readable_text
- [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.

# data[].attributes.attachment_link.title
Introduction to powers of 10 (video) | Khan Academy

# data[].attributes.attachment_link.description
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.

# data[].attributes.attachment_link.readable_text
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.

Video transcript

what we're going to do in this video is refresh our understanding of place value but we're going to dig a little bit deeper and think about place value in the context of decimals so just as a refresher if I had the number 973 this should be review for you we already know that this rightmost space right over here this is the ones place and if we move one space to the left of that this is the tens place notice we went from ones to tens tens are ten times as much as ones and then we move one space to the left of that we go to the we multiply by tens again we get to the hundreds space and so this nine doesn't just represent nine it represents nine hundred or we could write that as nine hundred similarly the seven doesn't just represent seven it represents seven tens or 70 this 3 represents three one so it actually does represent three but as I promised we're now going to extend our end to understanding and what we do is we put a decimal here which you've probably seen before at the right and the way the reason why we even need a decimal is to really tell us where our ones place is we say okay if we go right to the left of the decimal that's going to be our one space because once we start introducing decimals we can introduce as many spaces as we want to the right of the decimal and so let's think about those a little bit if when we went from hundreds to tens notice we divided by ten when we go from tens to ones notice you divide by ten so what do you think this place over here is going to be called well what happens if you take 1/10 well then you get 1/10 so as you might imagine this is the tenths place and then if you were to go one place to the right of that what would this place be what would be tenths divided by ten or one tenth of a tenth so this would be a hundred hundreds place and then if you were to go one space to the right and we could keep doing this forever but if we were to go one space to the right of that what would it be well a hundreth divided by ten or one tenth of a hundredth is a thousand thousand space and so for example if I were to extend this number instead of it just being 973 if I were to write nine hundred seventy three point five two six what do these numbers these digits represent this five doesn't just represent five it represents it represents five tenths or another way of writing five tenths you could write it like this zero point five you just have a five in the tenths place or you could write it as five tenths this two and I think you get where this is going this doesn't just represent two it represents two hundredths I'm just going to make it very explicit in this video so it's very clear two hundredths another way to write that is you just write a two in the hundredths place so we're going one two spaces to the right of the decimal or you could write it as two over a hundred two hundredths and so you know for kicks pause the video what are all the different ways of representing this six what does this six are present well this is six thousand six thousand there's thousands there you go how would I could also write that a zero point let's use the tenths place hundredths place and then in the thousandths place I have six or I could write this as 6 over one thousand six thousands so big picture place value we can keep going to the right of the decimal and we can start representing things that are I guess you could say more precise

# data[].attributes.attachment_link.title
Place value with decimals (video) | Khan Academy

# data[].attributes.attachment_link.description
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.

# data[].attributes.attachment_link.title
Worked example: Write decimal in expanded form (video) | Khan Academy

# data[].attributes.attachment_link.description
Learn to write 905.074 in expanded form.

# data[].attributes.attachment_link.title
Comparing decimals: 9.97 and 9.798 (video) | Khan Academy

# data[].attributes.attachment_link.description
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.

# data[].attributes.attachment_link.readable_text
We're asked to compare 156.378 to 156.348. So I assume that they're asking us which of these is greater. And so let's rewrite these numbers. 156.378, and we want to compare that to 156.348. And it might already be obvious to you which of these is greater. But I'll work through it step-by-step, just in case it's not. So let's first go to the hundreds place. And you could view this as, to some degree, the most significant place, because this is where they have the largest-- the 1 in the hundreds place represents a lot more than a 1 in the tens place or a 1 in the six's place. A 1 in the hundreds place represents 100. It's this many hundreds. But we have the same number of hundreds in both situations. We have 1 hundred and we have 1 hundred. Let's go to the tens place. We have 5 tens here, which is 50. And we have 50 right over here. So, so far they're equal, so far, if you just look at the hundreds and the tens place. Let's go the ones place. We have a 6 here in the ones place, 6 ones, which is really just 6. And you also have 6 ones. So, so far, they're equal again. Let's go to the tenths place-- 3 tenths, 3 tenths. Let's go to the hundredths place. So we're going into smaller and smaller places. So then you have the 7 hundredths. And now we have 4 hundredths here. So this has less hundredths than this does over here. And if we go to the thousandths place, these are equal. But it doesn't matter, even if these were unequal. It doesn't matter, even if this one was larger than this one, because in the more significant place, this number was larger than this number. So 7 is greater than 4. We have 7 hundredths here. We have 4 hundredths here. So we get this number is greater than this number down here. Or 156.378 is greater than 156.348. And in particular, it's because of the hundredths place.

# data[].attributes.attachment_link.title
Comparing decimals: 156.378 and 156.348 (video) | Khan Academy

# data[].attributes.attachment_link.description
We're asked to compare 156.378 to 156.348. So I assume that they're asking us which of these is greater. And so let's rewrite these numbers. 156.378, and we want to compare that to 156.348. And it might already be obvious to you which of these is greater.

# data[].attributes.attachment_link.title
UNIT 5 STUDY GUIDE_ PLACE VALUE PATTERNS ONLY