求$$$\sqrt[3]{8}$$$

求$$$\sqrt[3]{8}$$$。

$$$8$$$ 的极坐标形式是 $$$8 \left(\cos{\left(0 \right)} + i \sin{\left(0 \right)}\right)$$$（步骤请参见 极坐标形式计算器）。

根据棣莫弗公式，复数$$$r \left(\cos{\left(\theta \right)} + i \sin{\left(\theta \right)}\right)$$$的所有$$$n$$$次方根由$$$r^{\frac{1}{n}} \left(\cos{\left(\frac{\theta + 2 \pi k}{n} \right)} + i \sin{\left(\frac{\theta + 2 \pi k}{n} \right)}\right)$$$, $$$k=\overline{0..n-1}$$$给出。

我们有$$$r = 8$$$、$$$\theta = 0$$$和$$$n = 3$$$。

• $$$k = 0$$$: $$$\sqrt[3]{8} \left(\cos{\left(\frac{0 + 2\cdot \pi\cdot 0}{3} \right)} + i \sin{\left(\frac{0 + 2\cdot \pi\cdot 0}{3} \right)}\right) = 2 \left(\cos{\left(0 \right)} + i \sin{\left(0 \right)}\right) = 2$$$
• $$$k = 1$$$: $$$\sqrt[3]{8} \left(\cos{\left(\frac{0 + 2\cdot \pi\cdot 1}{3} \right)} + i \sin{\left(\frac{0 + 2\cdot \pi\cdot 1}{3} \right)}\right) = 2 \left(\cos{\left(\frac{2 \pi}{3} \right)} + i \sin{\left(\frac{2 \pi}{3} \right)}\right) = -1 + \sqrt{3} i$$$
• $$$k = 2$$$: $$$\sqrt[3]{8} \left(\cos{\left(\frac{0 + 2\cdot \pi\cdot 2}{3} \right)} + i \sin{\left(\frac{0 + 2\cdot \pi\cdot 2}{3} \right)}\right) = 2 \left(\cos{\left(\frac{4 \pi}{3} \right)} + i \sin{\left(\frac{4 \pi}{3} \right)}\right) = -1 - \sqrt{3} i$$$

$$$\sqrt[3]{8} = 2$$$A

$$$\sqrt[3]{8} = -1 + \sqrt{3} i\approx -1 + 1.732050807568877 i$$$A

$$$\sqrt[3]{8} = -1 - \sqrt{3} i\approx -1 - 1.732050807568877 i$$$A