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# Scientific notation examples

CCSS.Math:

## Video transcript

there are two whole Khan Academy videos on what scientific notation is why we even worry about it and also goes through a few examples so what I want to do in this video is just use the ck-12 org algebra one book to do some more to do some more scientific notation examples so let's take some things that are written in scientific notation just as a reminder scientific notation is useful because it allows us to write really large or really small numbers in ways that are easy for our brains to one write down and to understand so let's write down some numbers so let's say I have 3 point 1 0 2 times 10 to the 2nd and I want to write it in as just a numerical value it's in scientific notation already it's written as a product with a with a power of 10 so how do I write this it's just a numeral well there's a slow way in the fast way the slow way is to say well this is the same thing as three point one zero two times 100 which means if you multiply three point one zero two times 100 it'll be three one zero two with two zeros behind it and then we have one two three numbers behind the decimal point one two three numbers behind the decimal point and that would be the right answer this is equal to three hundred and ten point two now a faster way to do this is just to say well look right now I have only the three in front of the decimal point when I take something to the second times 10 to the second power I'm essentially shifting the decimal point two to the right so three point one zero two times 10 to the second power is the same thing as if I shift the decimal point one and then two because this is ten to the second power it's the same thing as three hundred and ten point two so this might be a faster way of viewing it every time you multiply it by ten you shift the decimal to the right by one let's do another example let's say I had seven point four times ten to the fourth well let's just do this the fast way let's shift the decimal four to the right so seven point four times ten to the fourth times ten to the one you're going to get 74 then times 10 to the second you're going to get 740 we're going to have to add a zero there because we have to shift the decimal again 10 to the third you're going to have seven thousand four hundred and then 10 to the fourth you're going to have 74,000 notice I just took this decimal went one two three four spaces four spaces so this is equal to 74,000 and when I hat when I had 74 and had to shift the decimal one more to the right I had to throw a zero here I'm multiplying it by 10 another way to think about it is I need 10 spaces between the decimal or batur sorry between the leading digit and the decimal so right here I only have one space I'll need four spaces so one two three four let's do a few more examples cuz I think the more examples the more you'll get what's going on so I have 1.75 times 10 to the negative 3 this is in scientific notation and I want to just write the numerical value of this so when you take something to the negative times 10 to the negative power you shift the decimal to the left so this is 1.75 so if you do it times 10 to the negative one power you will go one to the left but if you do times 10 to the negative 2 power you'll go 2 to the left and you'd have to put a 0 here and then if you do times 10 to the negative 3 you'd go 3 to the left and you would have to add another 0 so you take this decimal and go 1 2 3 to the left so our answer would be 0.001 7 5 is the same thing as 1 point 7 5 times 10 to the negative 3 and another way to check that you got the right answer is is if you have a 1 right here if you count the one one including the zeros to the right of the decimal should be the same as the negative exponent here so you have one two three numbers behind the decimal so you should have that's the same thing as to the negative three power you're doing 1,000 so this is one thousandth right there let's do another example actually let's mix it up let's start with something that's written as a numeral and then write it in scientific notation so let's say I have one hundred and twenty thousand so that's just this numerical value and I want to write it in scientific notation so this I can write as I take the leading digit 1.2 times 10 to the and I just count how many digits there are behind the leading digit one two three four five so 1.2 times 10 to the fifth and if you want to handle internalize why that makes sense 10 to the fifth is 10,000 so one point - sorry one point - 10 to the fifth is a hundred thousand so it's one point two times one one two three four five you have five zeroes it's ten to the fifth so one point two times 100,000 is going to be a hundred twenty thousand it's going to be one in one-fifth times a hundred thousand so 120s hopefully that's sinking in so let's do another one let's say the numerical value is 1 million seven hundred sixty-five thousand two hundred forty-four I want to write this in scientific notation so I take the leading digit 1 put a decimal sign everything else goes behind the decimal seven six five two four four and then you count how many digits there were between the leading digit and I guess you could imagine the first decimal sign because you could have deaths you could have numbers that keep go over here so between the leading digit and the decimal sign and you have one two three four five six digits since this times 10 to the sixth and 10 to the six is just a million so it's one point seven six five two four four times a million which makes sense roughly 1.7 times a million is roughly 1.7 million this is you know a little bit more than 1.7 million so it makes sense let's do another one how do I write twelve in scientific notation same drill is equal to 1.2 times well we only have one digit between the one and the decimal spot or the decimal point so it's 1.2 times 10 to the first power or 1.2 times 10 which is definitely equal to 12 let's do a couple of examples where we're taking 10 to a negative power so let's say we had zero point zero zero two eight one and we want to write this in scientific notation so what you do is you just have to think well how many how many how many digits are there to get to the include the leading the leading numeral in the value so what I mean there is count 1 2 3 so what we want to do is we move the decimal one two three spaces so one way you could think about it is you could multiply to move the decimal to the right three spaces you would multiply it by 10 to the third but if you're multiplying something by 10 to the third you're changing its value so you also have to multiply by 10 to the negative 3 only this way will you not change the value right if I multiply by 10 to the 3 times 10 to the negative 3 3 minus 3 is 0 this is just like multiplying it by 1 so what is this going to equal if I take the decimal I move it 3 spaces to the right this part right here is going to be equal to 2 point 8 1 and then we're left with this 1 times 10 to the negative 3 now a very quick way to do it is just to say look let me count including the leading numeral how many spaces I have behind the decimal 1 2 3 so it's going to be 2.8 1 times 10 to the negative 1/2 3 power let's do one more like that let me actually scroll up here let's do one more like that let's say I have zero point let's say I have let's say 1 2 3 4 5 6 how many zeros do I have in this problem well I'll just make up something 0 2 7 and you wanted to write that in scientific notation well you count all the digits up to the 2 behind the decimal so 1 2 3 four five six seven eight so this is going to be two point seven times ten to the negative eight power now let's do another one where we start with the scientific notation value and we want to go to the numeric value just to mix things up so let's say you have 2.9 times 10 to the negative fifth so one way to think about is this leading numeral plus all the zeros to the left of the decimal spot is going to be five digits so you have a two and a nine and then you're going to have five and then you're gonna have four more zeros one two three four and then you're going to have your decimal and how did I say no Forde zeros because I'm counting this as one two three four five spaces behind the decimal including the leading numeral and so it's zero point zero zero zero zero to nine and just to verify do the other technique how do I write this in scientific notation I count the zero I count all of the digits all of the leading zeros behind the decimal including the leading non-zero numeral so I have one two three four five digits so it's ten to the negative five so it'll be two point nine two point nine times ten to the negative five and once again I want it you know this isn't just some type of black magic here this is this actually makes a lot of sense if I wanted to get this number to two point nine what I would have to do is move the decimal over one two three four five spots like that and to multiple to get something the decimal to move over to the right by five spots I'll have to say say with zero zero zero zero to nine if I multiply it by 10 to the fifth I'm also going to have to multiply it by ten to the negative five because I don't to change the number this right here is just multiplying something by one ten to the fifth times 10 to the negative five is one so this right here this part right here is essentially going to move the desk five to the right one two three four five so this will be two point five and then we're going to be left with times ten to the negative five anyway hopefully you found that scientific notation drill useful