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CCSS.Math:

Xpress zero point zero zero zero zero zero zero zero zero zero three four five seven in scientific notation so let's just remind ourselves what it means to be in scientific notation scientific notation will be some number times some power of ten where this number right here is going to be let me write it this way it's going to be greater than or equal to one and it's going to be less than ten so over here what we want to put here is that leading number is going to be and in general you're going to look for the first non-zero digit and this is the number that you're going to want to start off with this is the only number you're going to want to put ahead of or I guess it's to the left of the decimal point so we could write three point four five seven three point four five seven and it's going to be multiplied by ten to something now let's think about what we're going to have to multiply it by to go from three point four five seven to this very very small number I mean we have had to move the decimal from three point four five seven to get to this you have to move the decimal to the left a bunch you have to add a bunch of zeros to the left of the three you have to keep you have to keep moving the decimal over to the left to do that we're essentially making the number much much much smaller so we're going to we're not going to multiply it by a positive exponent of 10 we're going to multiply it times a negative exponent of 10 that the equivalent is you're kind of dividing by a positive exponent of 10 and so the best way to think about it when you move when you move you're an exponent one to the left you're dividing by ten which is equivalent to multiplying by 10 to the negative one power if your let me let me give you an example here so if I have 1 times 10 is clearly just equal to 10 one times 10 to the negative one that's equal to one times one over 10 which is equal to one over ten one times and let me actually write a decimal which is equal to zero which is equal to let me actually I skipped a step right there let me add one times 10 to the 0 so we have something so this is 1 times 10 to the first 1 times 10 to the 0 is equal to 1 times 1 which is equal to 1 1 times 10 to the negative 1 is equal to 1 over 10 which is equal to 0.1 if I do 1 times 10 to the negative 2 10 to the negative 2 is 1 over 10 squared or 1 over 100 so this is going to be 1 over 100 which is 0.01 what's happening here when I raise it to a negative power I raise it to a negative 1 power I've essentially moved the decimal from to the right of the one to the left of the one I moved from there to there when I raise it to the negative 2 I moved it two over to the left so how many how many times we're going to have to move it over to the left to get this number right over here so we essentially so let's think about how many zeros we have so we have to move it one time just to get in front of the 3 and then we have to move it that many more times to get all of the zeros in there so that we have to move it one time to get the 3 so if we started here we're going to move 1 2 3 4 5 6 7 8 9 10 times so this is going to be 3 point 4 5 7 times 10 to the negative 10 power let me just rewrite it so 3 point 4 5 7 times 10 to the negative 10 power so in general what you want to do is you want to find the first nonzero number here remember you want a number here that's between 1 and 10 and it can be equal to 1 but it has to be less than 10 3 point 4 5 7 definitely fits that bill it's between 1 and 10 and then you just want to count the leading zeros between the decimal and that number and include the number because that tells you how many times you have to shift the decimal over to actually get this number up here and so we have to shift this decimal 10 times to the left to get this thing up here