ivan77 Posted March 30, 2005 Share Posted March 30, 2005 I have been wondering how this is worked out without just plugging it into the calculator, is it possible to do it on paper or does it take a long time to work out? I feel i should already know this but i have never learned it in school or in my maths course. Its obviously not a ratio as: 9^1.5=27 but 9^2= 81. Link to comment Share on other sites More sharing options...
elfstone Posted March 30, 2005 Share Posted March 30, 2005 Well, if the decimal number is easy i know how to do it. A power of 0.5 equals square root, so 9^1.5 = 9^1.0 x 9^0.5 = 9x3 = 27. A power of 0.33 i suppose is the cubic root. I don't know how to generalize this though. Link to comment Share on other sites More sharing options...
Dave Posted March 30, 2005 Share Posted March 30, 2005 The way to approximate them is to use Taylor expansions, but I daresay you probably haven't done those yet. So, in a word, probably not Link to comment Share on other sites More sharing options...
ivan77 Posted March 31, 2005 Author Share Posted March 31, 2005 Are taylor expansions similiar to those used to find natural logs ie: e^2 for example. Link to comment Share on other sites More sharing options...
Dave Posted March 31, 2005 Share Posted March 31, 2005 You might have seen the formula: [math]e^x = 1 + x + \frac{1}{2!}x^2 + \frac{1}{3!}x^3 + \cdots[/math] This is the Taylor expansion for ex. Maybe my answer isn't quite correct; you can also use the binomial theorem to expand something like [math](1+x)^{1/2}[/math] and then get a pretty good approximation from that. Link to comment Share on other sites More sharing options...
ydoaPs Posted March 31, 2005 Share Posted March 31, 2005 how bout [math]a^\frac{b}{c}=(\sqrt[c]{a})^b[/math]? doesn't that work? Link to comment Share on other sites More sharing options...
Zeo Posted March 31, 2005 Share Posted March 31, 2005 I was going to say try using logarithms. We just learned about it last week in Pre-Cal . . . so maybe I'm not entirely sure what I'm talking about but . . . what's your problem anyway? Link to comment Share on other sites More sharing options...
Dave Posted March 31, 2005 Share Posted March 31, 2005 Not if b/c is, say, 1/2 and a is something nasty like 23. You can guesstimate, but it's a lot nicer if you use something like the binomial formula because you end up with putting decimals into polynomials, which is a lot easier. Link to comment Share on other sites More sharing options...
ydoaPs Posted March 31, 2005 Share Posted March 31, 2005 good point. i guess ur right. Link to comment Share on other sites More sharing options...
Dave Posted April 1, 2005 Share Posted April 1, 2005 It's the way most calculators work things like this out (or so I'm told). It's pretty accurate as well. Link to comment Share on other sites More sharing options...
Ducky Havok Posted April 1, 2005 Share Posted April 1, 2005 Not if b/c is, say, 1/2 and a is something nasty like 23. You can guesstimate, but it's a lot nicer if you use something like the binomial formula because you end up with putting decimals into polynomials, which is a lot easier. you could still use differentials to approximate it pretty close though. Like [math]\sqrt{23}[/math], you could say [math]x=25, dx=-2[/math], then [math]\frac{1}{2\sqrt{x}}dx+ \sqrt{x}[/math], which would give you [math]-1/5+5 [/math], which is 4.8. That's pretty close to the square root of 23. I'm only saying this because I don't understand how to use the binomial formula though. Can you please explain it some? (I might know it, just not know that that's what it is. Most likely I don't though) Link to comment Share on other sites More sharing options...
Dave Posted April 1, 2005 Share Posted April 1, 2005 Ack! For my perspective on seperating dx's and all of that, have a look at the "dt ramblings" thread in the Calculus forum But yeah, that's just guesstimation really. Well, the binomial formula is this: [math](1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!} x^3 \cdots[/math] I'm pretty sure you know that formula Link to comment Share on other sites More sharing options...
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