Buscar $$$\sqrt[3]{-8}$$$

Encuentra $$$\sqrt[3]{-8}$$$.

La forma polar de $$$-8$$$ es $$$8 \left(\cos{\left(\pi \right)} + i \sin{\left(\pi \right)}\right)$$$ (para conocer los pasos, consulte calculadora de forma polar).

Según la fórmula de De Moivre, todas las raíces $$$n$$$ -ésimas de un número complejo $$$r \left(\cos{\left(\theta \right)} + i \sin{\left(\theta \right)}\right)$$$ están dadas por $$$r^{\frac{1}{n}} \left(\cos{\left(\frac{\theta + 2 \pi k}{n} \right)} + i \sin{\left(\frac{\theta + 2 \pi k}{n} \right)}\right)$$$, $$$k=\overline{0..n-1}$$$.

Tenemos que $$$r = 8$$$, $$$\theta = \pi$$$ y $$$n = 3$$$.

• $$$k = 0$$$: $$$\sqrt[3]{8} \left(\cos{\left(\frac{\pi + 2\cdot \pi\cdot 0}{3} \right)} + i \sin{\left(\frac{\pi + 2\cdot \pi\cdot 0}{3} \right)}\right) = 2 \left(\cos{\left(\frac{\pi}{3} \right)} + i \sin{\left(\frac{\pi}{3} \right)}\right) = 1 + \sqrt{3} i$$$
• $$$k = 1$$$: $$$\sqrt[3]{8} \left(\cos{\left(\frac{\pi + 2\cdot \pi\cdot 1}{3} \right)} + i \sin{\left(\frac{\pi + 2\cdot \pi\cdot 1}{3} \right)}\right) = 2 \left(\cos{\left(\pi \right)} + i \sin{\left(\pi \right)}\right) = -2$$$
• $$$k = 2$$$: $$$\sqrt[3]{8} \left(\cos{\left(\frac{\pi + 2\cdot \pi\cdot 2}{3} \right)} + i \sin{\left(\frac{\pi + 2\cdot \pi\cdot 2}{3} \right)}\right) = 2 \left(\cos{\left(\frac{5 \pi}{3} \right)} + i \sin{\left(\frac{5 \pi}{3} \right)}\right) = 1 - \sqrt{3} i$$$

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

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

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