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### Unit 14: Lesson 52

Multiplying and dividing with powers of 10- Exponents and powers of 10 patterns
- Multiplying a decimal by a power of 10
- Dividing a decimal by a power of 10
- Dividing a decimal by a power of 10: pattern
- Multiply and divide by powers of 10
- Understanding moving the decimal
- Fractions as division by power of 10

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# Exponents and powers of 10 patterns

CCSS.Math:

This lesson explains calculations involving powers of ten. It demonstrates how to multiply and divide by these powers, and explains the results of these calculations, such as adding or removing zeroes from the end of a number. Additionally, the lesson discusses how these calculations can shift the placement of a decimal point to the left or right. Created by Sal Khan.

## Video transcript

We are asked what is 10 to
the fifth power equivalent to. Well, 10 to the fifth power is
the same thing as taking a 1 and multiplying it
by 10 five times. So let's do that. So that's three, four, and five. So it's 1 times 10 times 10
times 10 times 10 times 10. Notice one, two,
three, four, five. And what's this going to be? Well, 1 times 10 is 10. 10 times 10 is 100. 100 times 10 is 1,000. 1,000 times 10 is 10,000. 10,000 times 10 is 100,000. So this is going to be 100,000. Now, you might have noticed,
every time we multiply it by 10, we're adding
another 0 to the product. So if we're multiplying
by 10 five times, we're going to add five
0's to the product. So this is literally going to
be 1 followed by five 0's, one, two, three, four, five. So 10 to the fifth is the
same thing as 100,000. Let's do another one that
is on a similar topic. How many 0's does the product
67 times 10 to the fifth have? Well, there's a bunch of
ways of thinking about that. 67 times 10 to the fifth
is the exact same thing, this is equivalent to-- you
could view this as 67 times. And we could use
the same exact logic that we just saw in
the previous problem. So 67 times 1 times 10 times
10 times 10 times 10 times 10. And we already figured
out exactly what this is. If you have 1 multiplied
by 10 five times, this right over here is going
to be 1 followed by five 0's. One, two, three, four, five. Or it's going to be 100,000. So there's a couple
ways to think about it. If you're looking at just
this product right over here, you could say, well, 67 times
1 is just going to be 67. And then every time
you multiply by 10, you're going to add another 0. So you could say, well,
this is just going to be 67. And we're going to be
multiplying by 10 five times, so we're going to
add five 0's here. One, two, three, four, five. And so this would
come to 6,700,000. Another way you could
think about it-- this is the same thing
as 67 times 100,000, so it's going to be 67, and
then we have five 0's here. So once again, one,
two, three, four, five. You get the exact same value. Let's do another one. How many 0's does the
quotient 5,700,000 divided by 10 to the third power have? Well, 10 to the third
power we already know. 10 to the third power-- that's
the same thing as 1 times 10 times 10 times 10, which
is the same thing as 1,000. And so if we're dividing by
that, that's the same thing. So another way of writing this
expression right over here. So 5,700,000 divided
by 10 to the third is the same thing as--
I could write this way. 5,700,000 divided by
10 times 10 times 10. We could put a 1 out here,
but this won't really change the value, which is
equivalent to 5,700,000 divided by 1,000. And either way you
think about it, every time you divide by 10,
you're going to eliminate one of these 0's. So if you divide
by 10 three times-- so if you divide by 10 once,
you're going to eliminate a 0. Divide by 10 again, you're
going to eliminate another 0. Divide by 10 again, you're
going to eliminate another 0. You're going to be
left with 5,700. Another way of thinking
about it is, well, if I'm dividing by something
that has three 0's, I'm going to eliminate three 0's. So if I eliminate three
0's, I am left with 5,700. Now, just to be
clear, I could only do this because this was 1,000. If this was like
3,000 or something, I could think of this
as 3 times 1,000. And then I could only cancel
out the 1,000 with these 0's. But the 3 I would then
have to divide separately. But only because I'm straight
up dividing by 1,000. I'm dividing by a power of 10. I have three 0's here. I can cancel it out
with three 0's there. Let's do another one. So it asks you to do a few. And these are the
type of questions that you might see
in the exercise. When 72.1 is multiplied
by 10 to the third, the decimal point moves
blank places to the blank. And in the exercise, you
might see a dropdown here. So remember, if you're
multiplying by-- and this is essentially
10 to the third, so you're multiplying
by 1,000, you're going to get a larger number. You're not going to
get a smaller number. And so every time
you multiply by 10, your decimal point is
going to move to the right because you're getting larger. So we're going to move
the decimal point. So 72.1, if we
multiplied it by 10 once, then we would move the
decimal point over once. And we'd get 721,
which makes sense. 72 times 10 is 720. 72.1 times 10 should be 721. But if we want to multiply
it by 10 three times, we're going to move the
decimal place over not just once, but twice,
and three times. And you say wait, wait. How could I move it over? There's nothing here. Well, you throw a 0 in there. And so you're going
to get, it's going to be equal to-- so it's
going to be equal to 72,100. Now, they're not asking us that. They're just asking
how we would do it. And we saw the decimal
point moves three places to the right. And the big key
here is you're going to get a larger number when
you multiply by 10 three times, or multiply by 1,000. Let's do this one. When 56 is divided
by 10 to the third, the decimal point moves
blank places to the blank. Well, dividing by
10 to the third is the same thing as
dividing by 10 three times. And every time you
divide by 10, it's going to become
a smaller number. So 56, and you might say,
where is the decimal point? Well, there's implicitly a
decimal point right over there. You divide by 10 once, you're
going to get a smaller number. 56 is going to become
5 and something, so it's going to
become literally 5.6. Divide by 10 again, it's
going to become 0.56. Divide by 10 again,
you're like, wait. If I keep moving the decimal
place again to the left, what am I moving
it to the left of? Well, you could throw
a 0 right over here. And so if you move the decimal
point three places to the left, you are left with 0.056. And you might want to put a
0 out here just for clarity. But what we did here is we moved
the decimal point three places to the left in this situation. We got a smaller value.