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State Syrup Production (liters) Maine 1.10 ⋅ 10^6 New Hampshire 3.14 ⋅ 10^5 New York 9.65 ⋅ 10^5 Vermont 1.89 ⋅ 10^6 Order the states from least to greatest syrup production. New Hampshire, Maine, New York, Vermont New Hampshire, New York, Maine, Vermont Vermont, Maine, New York, New Hampshire Maine, Vermont, New Hampshire, New Yor
This is really an exercise in scientific notation. Take the first value for example (for Maine) : 1.10.10^6. This can also be written as 1.10 x 10 ^6 either, and is essentially a more convenient way of writing a very large number than writing a certain number of zeros. So, if we take 10^6 ("Ten to the power of 6"), this is 10 multiplied by itself 6 times. 10^6 = 10 x 10 x 10 x 10 x 10 x 10 = 1,000,000 Thus, 1.10 x 10^6 is essentially 1.10 multiplied by 1,000,000, which gives us 1,100,000. If it's a number x 10^5, then we are multiplying that number by 100,000, not 1,000,000 (essentially dropping a 0). Any number can be written in scientific notation, and it is also useful when trying to write very small numbers in a convenient fashion. So, to identify which number is biggest when written in this form, we need to consider: -The number multiplied to 10^n -The value of n (i.e. the power which 10 is raised to in the notation) The strategy I would use to tackle your question is as follows: 1.) Look at the value of n in each case which the 10 is raised to. If they are all the same, then we can proceed to look at the numbers in front of the 10^n, as they are all being multiplied by the same thing. If not, then we can change the way the scientific notation is written so that they are. In this case, the value of n is not the same in each scientific notation, as two have 10^6 and two have 10^5. We can convert the figures with 10^6 to an expression with 10^5 so that the four are now comparable (we could so it the other way around either, it would make no difference). If we consider 1.0 x 10^6, for a moment, this is 1,000,000. If we wanted to write a million in scientific notation which had 10^5, we would have to change the number in front from 1.0 to 10, as 10 x 10^5 = 10 x 100,000 = 1,000,000. So, in converting those from 10^6 to 10^5, we move the decimal point one place to the right to produce a larger number in front. If we were converting the other way around (i.e. 10^5 to 10^6), we would move the decimal point of the number in front one place to the left to produce a smaller value. 2.) Once we have each of the scientific notations written with 10 raised to the same power of n in each case, we can simply compare the numbers in front of this to order from smallest to greatest. As were now multiplying each of these by the same thing (100,000 or 10^5), we can ignore this part of the notation in finding our answer. Hope that helps! :)