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I know that $\left ( -1 \right )^{2/3}=\left ( \left ( -1 \right )^{2} \right )^{1/3}=1$

But Matlab computes this as $- 0.5 + 0.8660254038i$ a complex number.Why?

  • 2
    You could also say that, $\left ( -1 \right )^{2/3}=\left ( \left ( -1 \right )^{\frac{1}{3}} \right )^{2}$2012-10-17
  • 1
    You are both right. As a function of a real variable, $f(x)=x^\frac{2}{3}$ means $(\sqrt[3]{x})^2$ and $f(-1) = 1$. One of the complex roots of $-1$ is $\frac{1}{2} + \frac{\sqrt{3}}{2}i$ and its square is $-\frac{1}{2} + \frac{\sqrt{3}}{2}i$, which Matlab gave you (approximately).2012-10-17
  • 1
    Remember that in the complex numbers there are *three* third roots for $1$; and that $\sqrt[3]{-1}^2$ is not necessarily the same thing as $\sqrt[3]{(-1)^2}$ for complex numbers.2012-10-17

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