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What is the exact value of this expression? $$ \left ( -2 \sqrt 2 \right )^{2 \over 3} $$ Isn't $2$ one of the answer? Wolfram gets $-1+i \sqrt 3$. Is root multivariable function for complex number?

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    I think there are three roots. Wolfram Alpha may not give you all three roots though. To make sure Wolfram Alpha give you $2$, use parentheses like this: ((-2 * sqrt(2))^2)^(1/3)2012-08-23
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    Sure, $n$-th root of anything but $0$ is $n$-valued.2012-08-23
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    $(-2\sqrt2)^{2/3}=2$. See http://en.wikipedia.org/wiki/Exponentiation#Rational_exponents and http://en.wikipedia.org/wiki/Exponentiation#Failure_of_power_and_logarithm_identities2012-08-23

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Let $x = (-2 \sqrt 2)^{2 \over 3}$.

$x^3 = (-2\sqrt{2})^2 = 8$.

Let $\omega \neq 1$ be a root of $x^3 = 1$. Then, the roots of $x^3 = 8$ are $2, 2\omega, 2\omega^2$. Let's compute $\omega$.

$x^3 - 1 = (x - 1)(x^2 + x + 1)$.

Hence $\omega = \frac{-1 + i\sqrt{3}}{2}$ or $\frac{-1 - i\sqrt{3}}{2}$.

Hence the roots of $x^3 = 8$ are $2, -1 + i\sqrt{3}, -1 - i\sqrt{3}$.

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    if all those values are result of expression, then is $2 = -1 + i \sqrt 3$. I mean what is the exact value of that expression?2012-08-23
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    There are 3 exact values.2012-08-23
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    @user10254, obviously $\,2=-1+i\sqrt 3\,$ is false, as complex numbers are equal iff their real and imaginary are equal, corresp. As the answer shows, there are 3 "exact" values for that expression, and without any further conditions none is "more correct" or "better" than other.2012-08-23
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    @DonAntonio how can an expression have three values? is it multivariable function? can you refer some topic for me to study?2012-08-23
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    @user10254 , it's like the expression $\,4^{1/2}\,$: what is it? Certainly mathematicians have agreed to take the positive root, but this is mainly an agreement to make $\,f(x)=\sqrt x\,$ an actual function. When checking the possible *values* the above expression has we must state both $\,-2\,,\,2\,$ , and none of these values is more accurate or better than the other one.2012-08-23
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    @DonAntonio can you refer me something to read??2012-08-23
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    @MonkeyD.Luffy , any introductory text book in complex numbers must include this stuff. Try college algebra or stuff like that (though in some parts, like here, this is covered in high school), or simply google "complex numbers, roots of unit, roots of complex numbers " or something like that. There are millions of sites.2012-08-24