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CCSS.Math:

whereas to multiply 1.45 times ten to the eighth times nine point two times ten to the negative twelve times three point zero one times 10 to the negative fifth and express the product in both decimal and scientific notation so this is one point four five times ten to the eighth power times and I could just write the parenthesis again like this but I'm just going to write it as another multiplication times nine point two times ten to the negative twelve and then x times three point zero one times 10 to the negative fifth all this meant when I wrote these parentheses times y next to each other I'm just going to multiply this expression times this expression times this expression and since everything is involved multiplication it actually doesn't matter what order I multiply in and so with that in mind I can swap the order here this is going to be the same thing as one point four five that's that right there times nine point two times nine point two times three point zero one times three point zero one times ten to the eighth let me do that in that purple color times ten to the eighth so times ten to the eighth times ten to the negative twelve power ten to the negative twelve power times ten to the negative fifth power times ten to the negative fifth power and this is useful because now I have all of my powers of ten right over here I could parenthesis around that and I have all of my noun powers of ten right over there and so I can simplify it if I have I have the same base ten all right over here so I can add the exponents this is going to be 10 to the 8 minus 12 minus 5 power minus 5 power and then all of this on the left hand side let me get a calculator out I have one point four five you could do it by hand but this is a little bit faster less likely to make a careless mistake times nine point two times three point zero one which is equal to forty point one five three four so this is equal to forty point one five three four and of course this is going to be multiplied times ten to this thing and so if we simplify this exponent you get forty point one five three four times ten to the eighth minus 12 is negative four minus five is negative nine 10 to the negative nine power now you might be tempted to say that this is already in scientific notation because they have some number here times some power of 10 but this is not quite official scientific notation and that's because in order for it to be in scientific notation this number right over here has to be greater than or equal to 1 and less than 10 and this is obviously not less than ten essentially for it to be in scientific notation you want a non-zero digit right over here and then you want your decimal and then the rest of everything else so here and you want to non-zero single digit over here here we obviously have here we have two digits this is larger than 10 or this is greater than or equal to 10 you want this thing to be less than 10 and greater than or equal to 1 so the best way to do that is to write this thing right over here in scientific notation this is the same thing as 4 point 0 1 5 3 4 times 10 and one way to think about it is to go from 40 to 4 we had to move this decimal over to the left moving a decimal over to the left to go from 40 to 4 you're dividing by 10 so you have to multiply by 10 so it all equals out divided by 10 and then multiplied by 10 or another way to write it or another way to think about it is 4 point 0 and all this stuff times 10 is going to be 40 time point 1 5 3 4 and so you're going to have 4 all of this times 10 to the first power that's the same thing as 10 times this thing times 10 to the negative ninth power and then once again powers of 10 so it's 10 to the 1st times 10 to the negative 9 is going to be 10 to the negative 8 power 10 to the negative 8th power and we still have this four point zero one five three four times ten to the negative eight and now we have written it in scientific now we have written it in scientific notation now they want to just express it in both the decimal and the scientific notation and when they're asking us to write it in decimal notation they're essentially want us to multiply this out expand this out and so the way to think about it write these digits out so I have four zero one five three four and if I'm just looking at this number I start with the decimal right over here now every time every time I divide by ten or if I multiply by ten to the negative one I'm moving this over to the left one spot so 10 to the negative one if I multiply by 10 to the negative one that's the same thing as dividing by 10 and so I'm moving the decimal over to the left one here I'm multiplying by 10 to the negative 8 or you could say I'm dividing by 10 to the 8th power so I'm going to want to move the decimal to the left eight times so move move a decimal to left 8 eight times and one way to remember it look this is a very very very very small number if I multiply this I should get a smaller number so I should be moving the decimal to the left if this was a positive eight then this would be a very large number and so if I multiply by a large power of 10 I'm going to be moving the decimal to the right so this is going to be this whole thing should be evaluate it should evaluate to being smaller than four point zero one five three four so I move the decimal eight times to the left eight times to the left I move it one time to the left to get it right over here and then the next seven times I'm just going to add zeros so one two three four five six seven zeros and I'll put a zero in front of the decimal just to clarify it so now I notice if you include this digit right over here I have a total of eight digits I have an eight I have I have eight this are I have seven zeros and this digit gives us a so again one two three four five six seven eight the best way to think about it is I started with the decimal right here I moved once twice three four five six seven eight times that's what multiplying times 10 to the negative 8 did for us and I get this number right over here and when you see a number like this you start to appreciate why we write things in scientific notation this is much easier to take less space to write and you immediately know roughly how big this number here this is much harder to write you might even forget a zero when you write it or you might add a zero and now the person has to sit and count the zeros to figure out essentially how large or get a rough sense of how large this this this thing is it's one two three four five six seven zeros and you have this digit right here that's what gets us to that eight but this is a much much more complicated looking number than the one in scientific notation