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There are two whole Khan Academy videos on what

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scientific notation is, why we even worry about it.

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And it also goes through a few examples.

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And what I want to do in this video is just use a ck12.org

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Algebra I book to do some more scientific notation examples.

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So let's take some things that are written

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in scientific notation.

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Just as a reminder, scientific notation is useful because it

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allows us to write really large, or really small

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numbers, in ways that are easy for our brains to, one, write

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down, and two, understand.

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So let's write down some numbers.

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So let's say I have 3.102 times 10 to the second.

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And I want to write it as just a numerical value.

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It's in scientific notation already.

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It's written as a product with a power of 10.

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So how do I write this?

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It's just a numeral.

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Well, there's a slow way and the fast way.

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The slow way is to say, well, this is the same thing as

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3.102 times 100, which means if you multiplied 3.102 times

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100, it'll be 3, 1, 0, 2, with two 0's behind it.

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And then we have 1, 2, 3 numbers behind the decimal

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point, and that'd be the right answer.

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This is equal to 310.2.

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Now, a faster way to do this is just to say, well, look,

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right now I have only the 3 in front of the decimal point.

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When I take something times 10 to the second power, I'm

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essentially shifting the decimal point 2 to the right.

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So 3.102 times 10 to the second power is the same thing

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as-- if I shift the decimal point 1, and then 2, because

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this is 10 to the second power-- it's

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same thing as 310.2.

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So this might be a faster way of doing it.

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Every time you multiply it by 10, you shift the decimal to

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the right by 1.

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Let's do another example.

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Let's say I had 7.4 times 10 to the fourth.

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Well, let's just do this the fast way.

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Let's shift the decimal 4 to the right.

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So 7.4 times 10 to the fourth.

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Times 10 to the 1, you're going to get 74.

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Then times 10 to the second, you're going to get 740.

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We're going to have to add a 0 there, because we have to

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shift the decimal again.

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10 to the third, you're going to have 7,400.

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And then 10 to the fourth, you're going to have 74,000.

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Notice, I just took this decimal and

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went 1, 2, 3, 4 spaces.

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So this is equal to 74,000.

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And when I had 74, and I had to shift the decimal 1 more to

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the right, I had to throw in a 0 here.

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I'm multiplying it by 10.

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Another way to think about it is, I need 10 spaces between

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the leading digit and the decimal.

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So right here, I only have 1 space.

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I'll need 4 spaces, So, 1, 2, 3, 4.

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Let's do a few more examples, because I think the more

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examples, the more you'll get what's going on.

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So I have 1.75 times 10 to the negative 3.

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This is in scientific notation, and I want to just

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write the numerical value of this.

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So when you take something to the negative times 10 to the

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negative power, you shift the decimal to the left.

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So this is 1.75.

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So if you do it times 10 to the negative 1 power, you'll

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go 1 to the left.

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But if you do times 10 to the negative 2 power, you'll go 2

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to the left.

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And you'd have to put a 0 here.

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And if you do times 10 to the negative 3, you'd go 3 to the

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left, and you would have to add another 0.

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So you take this decimal and go 1, 2, 3 to the left.

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So our answer would be 0.00175 is the same thing as 1.75

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times 10 to the negative 3.

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And another way to check that you got the right answer is if

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you have a 1 right here, if you count the 1, 1 including

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the 0's to the right of the decimal should be the same as

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the negative exponent here.

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So you have 1, 2, 3 numbers behind the decimal.

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That's the same thing as to the negative 3 power.

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You're doing 1,000th, so this is 1,000th right there.

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Let's do another example.

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Actually let's mix it up.

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Let's start with something that's written as a numeral

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and then write it in scientific notation.

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So let's say I have 120,000.

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So that's just its numerical value, and I want to write it

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in scientific notation.

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So this I can write as-- I take the leading digit-- 1.2

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times 10 to the-- and I just count how many digits there

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are behind the leading digit.

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1, 2, 3, 4, 5.

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So 1.2 times 10 to the fifth.

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And if you want to internalize why that makes sense, 10 to

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the fifth is 10,000.

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So 1.2-- 10 to the fifth is 100,000.

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So it's 1.2 times-- 1, 1, 2, 3, 4, 5.

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You have five 0's.

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That's 10 to the fifth.

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So 1.2 times 100,000 is going to be a 120,000.

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It's going to be 1 and 1/5 times 100,000, so 120.

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So hopefully that's sinking in.

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So let's do another one.

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Let's say the numerical value is 1,765,244.

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I want to write this in scientific notation, so I take

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the leading digit, 1, put a decimal sign.

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Everything else goes behind the decimal.

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7, 6, 5, 2, 4, 4.

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And then you count how many digits there were between the

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leading digit, and I guess, you could imagine, the first

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decimal sign.

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Because you could have numbers that keep going over here.

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So between the leading digit and the decimal sign.

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And you have 1, 2, 3, 4, 5, 6 digits.

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So this is times 10 to the sixth.

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And 10 to the sixth is just 1 million.

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So it's 1.765244 times 1 million, which makes sense.

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Roughly 1.7 times million is roughly 1.7 million.

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This is a little bit more than 1.7

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million, so it makes sense.

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Let's do another one.

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How do I write 12 in scientific notation?

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Same drill.

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It's equal to 1.2 times-- well, we only have 1 digit

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between the 1 and the decimal spot, or the decimal point.

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So it's 1.2 times 10 to the first power, or 1.2 times 10,

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which is definitely equal to 12.

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Let's do a couple of examples where we're taking 10 to a

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negative power.

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So let's say we had 0.00281, and we want to write this in

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scientific notation.

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So what you do, is you just have to think, well, how many

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digits are there to include the leading

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numeral in the value?

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So what I mean there is count, 1, 2, 3.

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So what we want to do is we move the

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decimal 1, 2, 3 spaces.

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So one way you could think about it is, you can multiply.

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To move the decimal to the right 3 spaces, you would

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multiply it by 10 to the third.

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But if you're multiplying something by 10 to the third,

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you're changing its values.

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So we also have to multiply by 10 to the negative 3.

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Only this way will you not change the value, right?

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If I multiply by 10 to the 3, times 10 to the negative 2-- 3

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minus 3 is 0-- this is just like multiplying it by 1.

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So what is this going to equal?

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If I take the decimal and I move it 3 spaces to the right,

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this part right here is going to be equal to 2.81.

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And then we're left with this one, times 10 to

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the negative 3.

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Now, a very quick way to do it is just to say, look, let me

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count-- including the leading numeral-- how many spaces I

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have behind the decimal.

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1, 2, 3.

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So it's going to be 2.81 times 10 to the

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negative 1, 2, 3 power.

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Let's do one more like that.

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Let me actually scroll up here.

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Let's do one more like that.

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Let's say I have 1, 2, 3, 4, 5, 6-- how many 0's do I have

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in this problem?

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Well, I'll just make up something.

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0, 2, 7.

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And you wanted to write that in scientific notation.

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Well, you count all the digits up to the

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2, behind the decimal.

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So 1, 2, 3, 4, 5, 6, 7, 8.

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So this is going to be 2.7 times 10 to

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the negative 8 power.

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Now let's do another one, where we start with the

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scientific notation value and we want to go

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to the numeric value.

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Just to mix things up.

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So let's say you have 2.9 times 10 to

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the negative fifth.

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So one way to think about is, this leading numeral, plus all

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0's to the left of the decimal spot, is

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going to be five digits.

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So you have a 2 and a 9, and then you're going

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to have 4 more 0's.

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1, 2, 3, 4.

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And then you're going to have your decimal.

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And how did I know 4 0's?

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Because I'm counting,, this is 1, 2, 3, 4, 5 spaces behind

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the decimal, including the leading numeral.

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And so it's 0.000029.

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And just to verify, do the other technique.

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How do I write this in scientific notation?

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I count all of the digits, all of the leading 0's behind the

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decimal, including the leading non-zero numeral.

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So I have 1, 2, 3, 4, 5 digits.

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So it's 10 to the negative 5.

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And so it'll be 2.9 times 10 to the negative 5.

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And once again, this isn't just some type

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of black magic here.

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This actually makes a lot of sense.

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If I wanted to get this number to 2.9, what I would have to

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do is move the decimal over 1, 2, 3, 4, 5 spots, like that.

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And to get the decimal to move over the right by 5 spots--

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let's just say with 0, 0, 0, 0, 2, 9.

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If I multiply it by 10 to the fifth, I'm also going to have

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to multiply it by 10 to the negative 5.

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So I don't want to change the number.

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This right here is just multiplying something by 1.

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10 to the fifth times 10 to the negative 5 is 1.

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So this right here is essentially going to move the

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decimal 5 to the right.

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1, 2, 3, 4, 5.

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So this will be 2.5, and then we're going to be left with

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times 10 to the negative 5.

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Anyway, hopefully, you found that scientific

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notation drill useful.

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