Dividing with PowersDate: 01/16/2003 at 21:17:20 From: John Doe Subject: Dividing 7 I am having trouble with problems like this 6.0 x 10 ---------- 2 3.0 x 10 Will you please help?!? I just don't know what to multiply or divide or add or subtract from what. Date: 01/17/2003 at 13:28:58 From: Doctor Ian Subject: Re: Dividing Hi John, Let's consider a slightly different problem. Suppose we wanted to divide 6 * 10 ------ 3 * 5 We could multiply everything out first to get 6 * 10 60 ------ = -- = 4 3 * 5 15 But we could also do this: 6 * 10 6 10 ------ = - * -- = 2 * 2 = 4 3 * 5 3 5 That is, the rule for multiplying fractions is a b a * b - * - = ----- c d c * d but we can use the same rule to combine fractions, or to break them up into separate fractions. If this doesn't make sense to you, please let me know and I'll try to find another way to explain it, because it's a very important idea. Now, let's look at your example, and use the same idea: 6.0 * 10^7 6.0 10^7 ---------- = --- * ---- 3.0 * 10^2 3.0 10^2 10^7 = 2 * ---- 10^2 So, what do we do with the powers of 10? There are a few ways to think about it. One is to just use the definition of exponents: 10^7 10 * 10 * 10 * 10 * 10 * 10 * 10 ---- = -------------------------------- 10^2 10 * 10 10 * 10 = -------- * (10 * 10 * 10 * 10 * 10) 10 * 10 = 1 * (10 * 10 * 10 * 10 * 10) = 10^5 All we did, really, was cancel out some of the 10's in the exponent. A quicker way to do that is by crossing out zeros: xx 10^7 10000000 ---- = -------- = 100000 = 10^5 10^2 100 xx (Note that with powers of 10, the exponent is the same as the number of zeros following the 1.) Do you see why it's really the same thing? There's an even quicker way to do it, which is to subtract the bottom exponent from the top one: 10^7 ---- = 10^(7-2) = 10^5 10^2 After you've been doing this a while, that's probably the method you'll eventually decide to use. But there's no need to rush. Anyway, the main idea is that you deal with the powers of 10 separately: 6.0 * 10^7 6.0 10^7 ---------- = --- * ---- = 2.0 * 10^5 3.0 * 10^2 3.0 10^2 This, by the way, is one of the reasons we use powers of 10. It's much easier to divide one small number by another, and deal with the exponents using subtraction, than it is to divide one big number by another big number. Does that make sense? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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