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Express 0.0000000003457 in scientific notation

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So let's just remind ourselves what it means to be in scientific notation

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Scientific notation will be some number times some power of 10

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Where this number right here is going to be

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Let me write it this way

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It's going to be greater than or equal to 1, and it's going to be less than 10

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So over here, what we want to put here is that leading number is going to be

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And in general you're going to look for the first non-zero digit

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And this is the number that you're going to want to start off with

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This is the only number you're going to want to put ahead of

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– or I guess to the left of –

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Of the decimal point

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So we could write 3.457

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3.457

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And it's going to be multiplied by 10 to something

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Now let's think about what we're going to have to multiply it by

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To go from 3.457 to this very, very small number

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I mean we have had to move the decimal from 3.457 to get to this

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You have to move the decimal to the left a bunch

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You have to add a bunch of 0's to the left of the 3

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You have to keep moving the decimal over to the left

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To do that, we're essentially making the number much, much, much smaller

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So we are not going to multiply it with a positive exponent of 10

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We are going to multiply it times a negative exponent of 10

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that the equivalent is you kind of dividing by a positive exponent of 10

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the best way to think about it

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When you move your exponent one to the left

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You dividing by 10 which is equivalent to multiply by 10 to the negative 1 power

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Let me give you an example here

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So if I have 1 times 10 is clearly just equal to 10

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1 times 10 to the negative 1

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That's equal to 1 times 1/10 which is equal to 1/10

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1 times---which is equal to 0--I skip the step right there

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Let me add 1 times 10 to the 0

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So we have something natural

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So this is 1 times to the first

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1 times to the 0 is equal to 1 times 1 which is equal to 1

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1 times 10 to the negative 1 is equal to 1/10 which is equal to 0.1

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If I do 1 times 10 to the negative 2

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10 to the negative 2 is 1/10 squared or 1/100

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so this is going to be 1/100 which is 0.01

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What happened here?

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When I read it to the negative power

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I read it to the negative 1 power

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I essentially move the decimal from the right of the one to the left of the one

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I move from there to there

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When I get to -2, I move the 2 over the left

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So how many times we are going to move it over to the left to get this number right over here

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So we essentially, so let's think about how many zeros we have

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So we have to move it 1 times to get in front of the 3

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and then we have to move it that means more times to get all of the zeros

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in there

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so we have to move it 1 time to get the 3

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So we start here

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We are going to move 1,2,3,4,5,6,7,8,9,10..10 times

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So this is going to be 3.457 times 10 to the negative 10 power

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Let me just rewrite it

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So 3.457 times 10 to the negative 10 power

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So in general, what you want to do is you want to find the first non zero number here

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Remember, you want a number here that between 1 and 10

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and it could be equal to 1 but it has to be less than 10

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3.457 definitely fits that bill

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It's between 1 and 10

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And then you just want to count the leading zero between the decimal of that number and

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and include the number because that tells you how many times you have to shift the decimal over to actually get this number

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up here

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So we have to shift this decimal 10 times to the left to get this thing up here