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I thought I knew root notation quite well, but now I am confused, and various sources say different things.

So how much is $\sqrt[3]{-27}$?

Wikipedia and google both clearly state that the solution would be $-3$ with this notation. However, it has complex principal root $1.5 + 2.59... * i$, which is the Wolfram alpha's solution.

And finally when I pluck it into JavaScript in form of

Math.pow(-27, 1/3)

it returns NaN, and Python throws math domain error.

What is going on? What does $\sqrt[3]{a}; a < 0$ actually mean? Is it really equivalent to ${a^\frac{1}{3}}$ ? And then there is the analogical question for $$\sqrt[n]{a^n}, a < 0; n\mod2 = 1$$ I am so confused by nth roots now.

BoltKey
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    Most of programming languages don't know how to make powers of negative base and float exponent, unless you use explicitly complex numbers. – ajotatxe Nov 19 '16 at 15:58
  • In most programming languages 1/3 won't be exactly equal to $\frac13$ so the actual root would be complex. – Ian Miller Nov 19 '16 at 16:00
  • Compare the inputs (-27)^(1/3) and cubroot(-27) in Wolfram alpha. It will depend on how their algorithm defines it.

    Personally, I would take the $\sqrt[3]{x}$ notation as the positive root.

     – tomcuchta Nov 19 '16 at 16:01
  • Well if you go historically and start with real numbers it denotes the real root. But in complex numbers the usual and more consistent convention is that the $\sqrt[3]{x}$ is actually the principal root (in this case not real). That's why these systems give you this answer. http://mathworld.wolfram.com/CubeRoot.html – Sil Nov 19 '16 at 16:09
  • Btw some related questions http://math.stackexchange.com/questions/1692727/, http://math.stackexchange.com/questions/25528/. – Sil Nov 19 '16 at 16:24
  • Also related. – Cameron Buie Jul 15 '18 at 21:14

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