Find $$$\sqrt[6]{-1000000}$$$

Find $$$\sqrt[6]{-1000000}$$$.

The polar form of $$$-1000000$$$ is $$$1000000 \left(\cos{\left(\pi \right)} + i \sin{\left(\pi \right)}\right)$$$ (for steps, see polar form calculator).

According to the De Moivre's Formula, all $$$n$$$-th roots of a complex number $$$r \left(\cos{\left(\theta \right)} + i \sin{\left(\theta \right)}\right)$$$ are given by $$$r^{\frac{1}{n}} \left(\cos{\left(\frac{\theta + 2 \pi k}{n} \right)} + i \sin{\left(\frac{\theta + 2 \pi k}{n} \right)}\right)$$$, $$$k=\overline{0..n-1}$$$.

We have that $$$r = 1000000$$$, $$$\theta = \pi$$$, and $$$n = 6$$$.

• $$$k = 0$$$: $$$\sqrt[6]{1000000} \left(\cos{\left(\frac{\pi + 2\cdot \pi\cdot 0}{6} \right)} + i \sin{\left(\frac{\pi + 2\cdot \pi\cdot 0}{6} \right)}\right) = 10 \left(\cos{\left(\frac{\pi}{6} \right)} + i \sin{\left(\frac{\pi}{6} \right)}\right) = 5 \sqrt{3} + 5 i$$$
• $$$k = 1$$$: $$$\sqrt[6]{1000000} \left(\cos{\left(\frac{\pi + 2\cdot \pi\cdot 1}{6} \right)} + i \sin{\left(\frac{\pi + 2\cdot \pi\cdot 1}{6} \right)}\right) = 10 \left(\cos{\left(\frac{\pi}{2} \right)} + i \sin{\left(\frac{\pi}{2} \right)}\right) = 10 i$$$
• $$$k = 2$$$: $$$\sqrt[6]{1000000} \left(\cos{\left(\frac{\pi + 2\cdot \pi\cdot 2}{6} \right)} + i \sin{\left(\frac{\pi + 2\cdot \pi\cdot 2}{6} \right)}\right) = 10 \left(\cos{\left(\frac{5 \pi}{6} \right)} + i \sin{\left(\frac{5 \pi}{6} \right)}\right) = - 5 \sqrt{3} + 5 i$$$
• $$$k = 3$$$: $$$\sqrt[6]{1000000} \left(\cos{\left(\frac{\pi + 2\cdot \pi\cdot 3}{6} \right)} + i \sin{\left(\frac{\pi + 2\cdot \pi\cdot 3}{6} \right)}\right) = 10 \left(\cos{\left(\frac{7 \pi}{6} \right)} + i \sin{\left(\frac{7 \pi}{6} \right)}\right) = - 5 \sqrt{3} - 5 i$$$
• $$$k = 4$$$: $$$\sqrt[6]{1000000} \left(\cos{\left(\frac{\pi + 2\cdot \pi\cdot 4}{6} \right)} + i \sin{\left(\frac{\pi + 2\cdot \pi\cdot 4}{6} \right)}\right) = 10 \left(\cos{\left(\frac{3 \pi}{2} \right)} + i \sin{\left(\frac{3 \pi}{2} \right)}\right) = - 10 i$$$
• $$$k = 5$$$: $$$\sqrt[6]{1000000} \left(\cos{\left(\frac{\pi + 2\cdot \pi\cdot 5}{6} \right)} + i \sin{\left(\frac{\pi + 2\cdot \pi\cdot 5}{6} \right)}\right) = 10 \left(\cos{\left(\frac{11 \pi}{6} \right)} + i \sin{\left(\frac{11 \pi}{6} \right)}\right) = 5 \sqrt{3} - 5 i$$$

$$$\sqrt[6]{-1000000} = 5 \sqrt{3} + 5 i\approx 8.660254037844386 + 5 i$$$A

$$$\sqrt[6]{-1000000} = 10 i$$$A

$$$\sqrt[6]{-1000000} = - 5 \sqrt{3} + 5 i\approx -8.660254037844386 + 5 i$$$A

$$$\sqrt[6]{-1000000} = - 5 \sqrt{3} - 5 i\approx -8.660254037844386 - 5 i$$$A

$$$\sqrt[6]{-1000000} = - 10 i$$$A

$$$\sqrt[6]{-1000000} = 5 \sqrt{3} - 5 i\approx 8.660254037844386 - 5 i$$$A