I've learnt that the square of any real number is always positive. So, we know $x^2\ge0$ for any real number $x$. Similarly, $x^{2k}\ge0$ for any $x\in\mathbb R$ and $k\in\mathbb N$. How can we find all of the forms of $r$ for which $x^r$ is positive if it is real too?
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Among the integer exponents, you've found them all (apart from $0$). – vadim123 Dec 15 '13 at 15:16
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if r mod 2 = 0 , then it will be positive – abkds Dec 15 '13 at 15:16
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I know, but the exponent shouldn't be necessarily an integer. – CODE Dec 15 '13 at 15:17
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If $x<0$, then $x^{k}$ is positive when $k$ is an even integer, negative when $k$ is an odd integer, and no longer real when $k$ is not an integer. – mjqxxxx Dec 15 '13 at 15:18
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@mjqxxxx But what about $(-1)^{\frac{2}{3}}$? – CODE Dec 15 '13 at 15:21
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if x>0 than r can be Any real number
if x<0 than Some of the systems are giving -1 and some of them are giving 1 for (-1)^(2/3) so it is ambiguous to give answer if You can clear than answer can be find out . You can check it on http://www.wolframalpha.com/input/?i=-1%5E%282%2F3%29&lk=4&num=2&lk=4&num=2 http://ideone.com/Mm3n9s
john
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No i think we can find all of the forms, as our teacher said too. + Please note that i want it to be positive IF REAL for all $x$. – CODE Dec 15 '13 at 16:56