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Scientific Notation - Problem 3
University of Michigan
Runs his own tutoring company
Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!
Scientific notation can make our life easier when we are multiplying really big and really small numbers. So what we want to do in this case is to rewrite all these numbers using scientific notation.
So figure out where we want our decimal spot and then count the number of digits we're moving our decimals over. So this is going to be 9, remember that a number has to be between 1 and 10 and now we're moving 1, 2, 3, 4, 5 so this is just 9 times 10 to the 5th.
So then times going across we want the decimal to be after the 4 because remember that our leading term that a has to be between 1 and 10, and then count the number of decimals we're moving 1, 2, 3, 4, 5, we're moving backwards so it's going to be a negative. Same thing for the bottom we end up with 2; 1, 2, 3, 4, 5, 6, it's going to be a positive 6 because we're moving to a big number and then our last one is 1 times and then 1, 2, 3, 4, 5, 6, 7, 8 and that is going to be a negative.
So what we have here are a number of things that we're multiplying together, but remember we can rearrange it when we're multiplying and dividing and then combine like terms. So with a little bit of rearranging I'm going to get all my standard numbers to one side and all my powers of 10 to the other, so what we end up with is this is equal to 9 times 4 times 10 to the 5th times 10 to the -5th over 2 times 1 times 10 to the 6th times 10 to the -8.
So what I'm going to do is first just focus on our powers of 10. Remember when we are multiplying powers, we can just add our exponents, so what we really end up with in the top is just 10 to the zero or just 1, these just cancel out and what we have down here adding these is just going to be 10 to the -2.
In the top we have 36 divide by 2 is 18, so what we really have then is 18 over 10 to the -2. Using our rules of exponents remember that a negative in the denominator becomes a positive if we flip it up, so this becomes 18 times 10² and the last thing to remember is that our first term has to be between 1 and 10. This is 18 so what I need to do is really move my decimal place over one unit so what I end up with is 1.8 times 10 to the third.
So what started out with as a pretty ugly multiplication, by rewriting everything in scientific notation and then just using what we know about combining like terms and laws of exponents, we were able to simplify it up quite easily.
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