Find all complex number solutions of the equation. $$ x^{3}+64 i=0 $$ Choose the correct solution set below. A. $$ \left\{4\left(\cos 90^{\circ}+i \sin 90^{\circ}\right), 4\left(\cos 210^{\circ}+i \sin 210^{\circ}\right), 4\left(\cos 330^{\circ}+i \sin 330^{\circ}\right)\right\} $$ B. $$ \left\{4\left(\cos 270^{\circ}+i \sin 270^{\circ}\right), 4\left(\cos 390^{\circ}+i \sin 390^{\circ}\right), 4\left(\cos 510^{\circ}+i \sin 510^{\circ}\right)\right\} $$ C. $$ \left\{64\left(\cos 90^{\circ}+i \sin 90^{\circ}\right), 64\left(\cos 210^{\circ}+i \sin 210^{\circ}\right)\right. $$, $$ \left.64\left(\cos 330^{\circ}+i \sin 330^{\circ}\right)\right\} $$ D. $$ \left\{64\left(\cos 270^{\circ}+i \sin 270^{\circ}\right), 64\left(\cos 390^{\circ}+i \sin 390^{\circ}\right)\right. $$, $$ \left.64\left(\cos 510^{\circ}+i \sin 510^{\circ}\right)\right\} $$

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Solve the equation $$ x^{3}+64i=0 $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$x^{3}+64i=0$$
- step1: Move the constant to the right side:
  $$x^{3}=0-64i$$
- step2: Remove 0:
  $$x^{3}=-64i$$
- step3: Simplify:
  $$x=\sqrt[3]{-64i}$$
- step4: Rewrite the complex number:
  $$x=\sqrt[3]{64\left(\cos\left(\frac{3\pi }{2}ight)+i\sin\left(\frac{3\pi }{2}ight)ight)}$$
- step5: Calculate the roots using a formula:
  $$x=\sqrt[3]{64}	imes \left(\cos\left(\frac{\frac{3\pi }{2}+2k\pi }{3}ight)+i\sin\left(\frac{\frac{3\pi }{2}+2k\pi }{3}ight)ight)$$
- step6: Simplify:
  $$x=4\left(\cos\left(\frac{\frac{3\pi }{2}+2k\pi }{3}ight)+i\sin\left(\frac{\frac{3\pi }{2}+2k\pi }{3}ight)ight)$$
- step7: Substitute $$k=0,1,2$$ in the expression$$:$$
  $$\begin{align}&x_{1}=4\left(\cos\left(\frac{\frac{3\pi }{2}+2	imes 0	imes \pi }{3}ight)+i\sin\left(\frac{\frac{3\pi }{2}+2	imes 0	imes \pi }{3}ight)ight)\&x_{2}=4\left(\cos\left(\frac{\frac{3\pi }{2}+2	imes 1	imes \pi }{3}ight)+i\sin\left(\frac{\frac{3\pi }{2}+2	imes 1	imes \pi }{3}ight)ight)\&x_{3}=4\left(\cos\left(\frac{\frac{3\pi }{2}+2	imes 2\pi }{3}ight)+i\sin\left(\frac{\frac{3\pi }{2}+2	imes 2\pi }{3}ight)ight)\end{align}$$
- step8: Calculate:
  $$\begin{align}&x_{1}=4\left(\cos\left(\frac{\pi }{2}ight)+i\sin\left(\frac{\pi }{2}ight)ight)\&x_{2}=4\left(\cos\left(\frac{7\pi }{6}ight)+i\sin\left(\frac{7\pi }{6}ight)ight)\&x_{3}=4\left(\cos\left(\frac{11\pi }{6}ight)+i\sin\left(\frac{11\pi }{6}ight)ight)\end{align}$$
- step9: Calculate:
  $$\begin{align}&x_{1}=4i\&x_{2}=-2\sqrt{3}-2i\&x_{3}=2\sqrt{3}-2i\end{align}$$
- step10: Rewrite:
  $$x_{1}=2\sqrt{3}-2i,x_{2}=-2\sqrt{3}-2i,x_{3}=4i$$
- step11: Remove the complex number(s):
  $$	extrm{No real solution}$$
The equation $$x^{3}+64i=0$$ does not have any real solutions.

Now, let's find the complex solutions of the equation $$x^{3}+64i=0$$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$x^{3}+64i=0$$
- step1: Move the constant to the right side:
  $$x^{3}=0-64i$$
- step2: Remove 0:
  $$x^{3}=-64i$$
- step3: Simplify:
  $$x=\sqrt[3]{-64i}$$
- step4: Rewrite the complex number:
  $$x=\sqrt[3]{64\left(\cos\left(\frac{3\pi }{2}ight)+i\sin\left(\frac{3\pi }{2}ight)ight)}$$
- step5: Calculate the roots using a formula:
  $$x=\sqrt[3]{64}	imes \left(\cos\left(\frac{\frac{3\pi }{2}+2k\pi }{3}ight)+i\sin\left(\frac{\frac{3\pi }{2}+2k\pi }{3}ight)ight)$$
- step6: Simplify:
  $$x=4\left(\cos\left(\frac{\frac{3\pi }{2}+2k\pi }{3}ight)+i\sin\left(\frac{\frac{3\pi }{2}+2k\pi }{3}ight)ight)$$
- step7: Substitute $$k=0,1,2$$ in the expression$$:$$
  $$\begin{align}&x_{1}=4\left(\cos\left(\frac{\frac{3\pi }{2}+2	imes 0	imes \pi }{3}ight)+i\sin\left(\frac{\frac{3\pi }{2}+2	imes 0	imes \pi }{3}ight)ight)\&x_{2}=4\left(\cos\left(\frac{\frac{3\pi }{2}+2	imes 1	imes \pi }{3}ight)+i\sin\left(\frac{\frac{3\pi }{2}+2	imes 1	imes \pi }{3}ight)ight)\&x_{3}=4\left(\cos\left(\frac{\frac{3\pi }{2}+2	imes 2\pi }{3}ight)+i\sin\left(\frac{\frac{3\pi }{2}+2	imes 2\pi }{3}ight)ight)\end{align}$$
- step8: Calculate:
  $$\begin{align}&x_{1}=4\left(\cos\left(\frac{\pi }{2}ight)+i\sin\left(\frac{\pi }{2}ight)ight)\&x_{2}=4\left(\cos\left(\frac{7\pi }{6}ight)+i\sin\left(\frac{7\pi }{6}ight)ight)\&x_{3}=4\left(\cos\left(\frac{11\pi }{6}ight)+i\sin\left(\frac{11\pi }{6}ight)ight)\end{align}$$
- step9: Calculate:
  $$\begin{align}&x_{1}=4i\&x_{2}=-2\sqrt{3}-2i\&x_{3}=2\sqrt{3}-2i\end{align}$$
- step10: Rewrite:
  $$x_{1}=2\sqrt{3}-2i,x_{2}=-2\sqrt{3}-2i,x_{3}=4i$$
The complex solutions of the equation $$x^{3}+64i=0$$ are $$x_{1}=2\sqrt{3}-2i$$, $$x_{2}=-2\sqrt{3}-2i$$, and $$x_{3}=4i$$.

Therefore, the correct solution set is:

A. $$ \left\{4\left(\cos 90^{\circ}+i \sin 90^{\circ}ight), 4\left(\cos 210^{\circ}+i \sin 210^{\circ}ight), 4\left(\cos 330^{\circ}+i \sin 330^{\circ}ight)ight\} $$
The correct solution set is A.

Find all complex number solutions of the equation.

Choose the correct solution set below.
A.
B.
C. ,

D. ,

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Find all complex number solutions of the equation. Choose the correct solution set below. A. B. C. , D. ,