方程全部根

写成复数形式：

$$64=2^6\,e^{i2\pi m},\quad m\in\mathbb{Z}.$$

取六次方根得

$$x=2\,e^{i\frac{2\pi m}{6}}=2e^{i\frac{m\pi}{3}},\quad m=0,1,2,3,4,5.$$

所以六个根为

$$\boxed{2,\ 1+i\sqrt3,\ -1+i\sqrt3,\ -2,\ -1-i\sqrt3,\ 1-i\sqrt3}.$$

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英文解析

Write in plural form:

$$64=2^6\,e^{i2\pi m},\quad m\in\mathbb{Z}.$$

Taking the sixth root yields

$$x=2\,e^{i\frac{2\pi m}{6}}=2e^{i\frac{m\pi}{3}},\quad m=0,1,2,3,4,5.$$

Thus, the six roots are

$$\boxed{2,\ 1+i\sqrt3,\ -1+i\sqrt3,\ -2,\ -1-i\sqrt3,\ 1-i\sqrt3}.$$