Resolving Decimal ExponentsDate: 03/26/2001 at 19:04:22 From: Adel Subject: Powers Please explain how to find 7^.3 or 5^.6 without using a calculator, and give me the formula. Thanks, Adel Date: 03/27/2001 at 12:58:58 From: Doctor Rick Subject: Re: Powers Hi, Adel. It's not easy, but we can at least think about how we might do it, and be glad we don't actually have to do it. Let's take 7^0.3 as our example. Write the decimal as a fraction: 7^(3/10) We can write 3/10 as 3 times 1/10. The multiplication property of exponents tells us that this can be written as: 7^(3 * 1/10) = (7^3)^(1/10) or 7^(1/10 * 3) = (7^(1/10))^3 Look at the second form: 7^(1/10) is the tenth root of 7. You can find methods for finding the square root and even the cube root of a number in the Dr. Math Archives, but they are methods of successive approximation. That is, you make a "guess" and use the method to find a better "guess," then use it again to find an even better "guess," etc. The number of steps it takes depends on how accurate you want your answer to be. Here is one approximation method (called Newton's Method) for finding the nth root of a number y: x_2 = x_1 * (1-1/n) + y/n/x_1^(n-1) You can start with any guess for x_1 - say, 1 - and this formula will give you a number x_2 that is a better approximation to the nth root of y. Then put this new number in place of x_1 in the same formula, and you'll get a third number that is an even better approximation. With n = 10 and y = 7, you'll get the answer 1.214814 (accurate to six decimal places) in seven steps. It's tedious without a calculator (I used a spreadsheet, which amounts to the same thing), but it can be done. Once you have the tenth root of 7, then you have to raise this result to the third power to get the answer to your problem. 1.214814 * 1.214814 * 1.214814 = 1.792789 Another approach is to use the properties of logarithms. In particular, log(7^0.3) = log(7) * 0.3 You can use printed log tables or a slide rule instead of a calculator to find the logarithm of 7, which is 0.84509804. Multiply this by 0.3 to get 0.253529412. Then you must use the log table in reverse, finding the number whose logarithm is 0.253529412. It turns out to be 1.79278996, which (as we just saw) is 7^(3/10). This method is easier than the first, but it does require something other than paper and pencil (and begs the question, "How are logarithms calculated?"). Anyway, those are two ways of calculating a number to a decimal power. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ |
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