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## Lesson 14: Multiplying, dividing, and estimating with scientific notation

# Multiplying in scientific notation example

## Video transcript

Multiply, expressing the
product in scientific notation. So let's multiply
first, and then let's try to get what we have
in scientific notation. Actually, before
we do that, let's just even remember what it means
to be in scientific notation. To be in scientific
notation-- and actually, each of these numbers right
here are in scientific notation. It's going to be
the form a times 10 to some power, where a can be
greater than or equal to 1, and it is going to
be less than 10. So both of these numbers are
greater than or equal to 1, and they are less
than 10, and they're being multiplied by
some power of 10. Let's see how we
could multiply this. So this over here, this is
just the exact same thing. So if I do this in magenta,
this is the exact same thing as 9.1 times 10 to the sixth
times 3.2 times-- actually, I don't have to write it. Let me write it all
with a dot notation to make it a little bit
more straightforward. I'm doing that in magenta. This is equal to 9.1
times 10 to the sixth-- let me do it in this green
color-- times 3.2 times 10 to the negative 5th power. Now in multiplication,
this comes from the associative property. It essentially allows us to
remove these parentheses. It says, look, you can
multiply like that first, or you could actually
multiply these guys first. You can reassociate them. And the commutative
property tells us that we can rearrange
this thing right here. What I want to
rearrange is I want to multiply the 9.1 times
the 3.2 first and then multiply that times 10 to the
sixth times 10 the negative 5. So I'm just going
to rearrange this using the commutative property. This is the same thing
as 9.1 times 3.2, and I'm going to reassociate. So I'm going to do these
first, and then that times 10 to the sixth times
10 to the negative 5. And the reason
why this is useful is that this is really
easy to multiply. We have the same
base here, base 10, and we're taking the product,
so we can add the exponents. So this part right over here,
10 to the sixth times 10 to the negative 5,
that's going to be 10 to the 6 minus 5 power,
or essentially just 10 to the first power, which
is really just equal to 10. And that's going to be
multiplied by 9.1 times 3.2. So let me do that over here. If I have 9.1 times
3.2, so at first I'm going to ignore the
decimal, so I'm just going to treat it
like 91 times 32. So I have 2 times 1 is 2. 2 times 9 is 18. I'll stick a 0 here because
I'm in the tens place now, multiplying everything
really by 30 not just by 3. That's why my zero is there. And I multiplied 3
times 1 to get 3, and then 3 times 9 is 27. And so it is 2. So I'm adding here. 2 plus 0 is 2. 8 plus 3 is 11, carry
or regroup that 1. 1 plus 1 is 2. 2 plus 7 is 9. And then I have a 2 here. So 91 times 32 is 2,912. But I didn't
multiply 91 times 32. I multiplied 9.1 times 3.2. So what I want to do
is count the number of digits I have behind
the decimal point. I have one, two digits
behind the decimal point, and so I'll have to have two
digits behind the decimal point in the answer. So one, two, I'll stick the
decimal right over there. This part right over here
comes out to be 29.12. You might feel like we're done. This kind of looks like
scientific notation. I have a number
times a power of 10. But remember, this number has to
be greater than or equal to 1-- which it is-- and less than 10. But this number is
not less than 10. It's not in scientific notation. What we can do is
let's just write this number in
scientific notation, and then we can use
the power of 10 part to multiply by this power of 10. 29.12, this is the
same thing as 2.912. Notice, what did I do to
go from there to there? I just moved the
decimal to the left. Or another way to
think about it, if I wanted to go from here to
there, what could I do to this? Well, I would multiply it by 10. If I multiplied it by 10,
I would move the decimal to the right. It would go from 2.9 to 29. So if I want to
write this value, this is just this times 10. So 29.12 is the same
thing as 2.912 times 10. Now, this is in
scientific notation, but that's just this part. And I still have to
multiply it by another 10, so times another 10. To finish up this problem, we
get 2.912 times 10 times 10, or 10 to the first
times 10 to the first. Well, what's that? Well, that's going to be
this part right over here. That's just 10 squared. So it's 2.912 times 10 to the
second power, and we are done.