Scientific notation
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Scientific Notation (old)
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Scientific Notation Examples
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Scientific notation intuition
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Scientific Notation
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Scientific Notation I
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Scientific Notation 3 (new)
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Scientific Notation Example 2
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Scientific notation
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Scientific notation 3
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Multiplying in Scientific Notation
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Multiplying and dividing scientific notation
Scientific Notation Example 2 Scientific Notation 2
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- Multiply, expressing the product in scientific notation.
- So let's multiply first,
- and then let's get what we have into scientific notation.
- Actually, before we do that,
- let's remember what it means to be in scientific notation.
- So, 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 ten to some power,
- where "a" can be greater than or equal to one,
- and it is going to be less than ten.
- So both of these numbers are greater than or equal to one,
- and less than ten,
- and they are being multiplied by some power of ten.
- Let's see how we can multiply this.
- So this over her is just the exact same thing as
- (I do this part in magenta)
- is the same as 9.1 times 10 to the sixth
- times ... (let me write it all out with a dot notation...
- make it a little more straight forward)
- So this is equal to 9.1 times 10 to the sixth
- times (in green) 3.2 times ten to the negative fifth power.
- Now, in multiplication, this comes from the associative property,
- which allows us to essentially remove these parentheses.
- It says, hey, you can multiply like that first,
- or you can actually multiply these guys first,
- you can re-associate them.
- And the commutative property tells us that
- we can rearrange this thing here.
- And what I want to rearrange is,
- I want to multiply the 9.1 times the 3.2 first,
- and then multiply that by 10 to the sixth times 10 to the negative five.
- So I'm just going to rearrange this using the commutative property.
- So this is the same thing as 9.1 times 3.2,
- and then I'm going to re-associate,
- so I'm going to do these first.
- And then that times
- 10 to the 6th times 10 to the negative 5.
- And the reason this is useful, is that this is really easy to multiply.
- We have the same base here, base 10,
- so we can, and we're taking the product,
- so we can add the exponents.
- So this part right over here,
- 10 to the 6th times 10 to the negative 5,
- that's going to be 10 to the 6 minus 5 power,
- so we're essentially just 10 to the first power,
- which is really just equal to ten.
- and that is going to be multiplied by 9.1 times 3.2
- Let me do that over here.
- 9.1 times 3.2. So first, I'm going to ignore the decimal.
- I'm just going to treat it like 91 times 32.
- So I have 2 times 1 is 2.
- 2 times 9 is 18.
- Take a zero her because I'm in the tenths place now,
- I'm really multiplying by 30,
- that's why my zero is there.
- I multiply 3 by 1 to get 3.
- And then 3 times 9 is 27.
- And so it is 2, (I'm adding here), 8 plus 3 is 11,
- carry/regroup the one, 1 plus 1 is two, 2 plus 7 is 9,
- and then I have a two here.
- 91 times 32 is 2912,
- but I didn't multiply 91 by 32 here,
- I multiplied 9.1 times 3.2.
- So I want to count the number of digits behind the decimal point.
- And 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.
- I'll stick the decimal right over there.
- So this part right over here,
- comes out to be 29.12.
- You might say, might feel like we're done.
- This kind of looks like scientific notation.
- I have a number times a power of ten.
- But remember,
- this number has to be greater than or equal to one,
- which it is, AND less than ten!
- But this number is not less than ten.
- Therefore, it is not scientific notation.
- So what we can do is, let's just write this number in
- scientific notation,
- and then we can use the multiply by ten part to multiply by
- this power of ten.
- So 29.12, this is the same thing as 2.912,
- notice, what did I do to get 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?
- I would multiply by ten.
- I would move the decimal to the right;
- It would go from 2.912 to 29.12.
- So if I want to write this value,
- this is just this [2.912] times ten.
- 29.12 is the same thing as 2.912 times 10.
- And now, this is in scientific notation,
- but that's just this part.
- We still have to multiply it by another 10.
- So, times another ten.
- And so, 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 just going to be 10 squared.
- So it is 2.912 times 10 to the second power.
- And we are done.
- Thanks You Sal! You are awesome! :)
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