Find all complex number solutions of the equation. $$ \mathrm{x}^{4}+256 i=0 $$ Choose the correct solution set below. A. $$ \left\{4\left[\cos \left(22.5^{\circ}\right)+i \sin \left(22.5^{\circ}\right)\right], 4\left[\cos \left(112.5^{\circ}\right)+i \sin \left(112.5^{\circ}\right)\right]\right. $$ $$ \left.4\left[\cos \left(202.5^{\circ}\right)+i \sin \left(202.5^{\circ}\right)\right], 4\left[\cos \left(292.5^{\circ}\right)+i \sin \left(292.5^{\circ}\right)\right]\right\} $$ B. $$ \left\{4\left[\cos \left(67.5^{\circ}\right)+i \sin \left(67.5^{\circ}\right)\right], 4\left[\cos \left(157.5^{\circ}\right)+i \sin \left(157.5^{\circ}\right)\right]\right. $$ $$ \left.4\left[\cos \left(247.5^{\circ}\right)+i \sin \left(247.5^{\circ}\right)\right], 4\left[\cos \left(337.5^{\circ}\right)+i \sin \left(337.5^{\circ}\right)\right]\right\} $$ C. $$ \left\{4\left[\cos \left(22.5^{\circ}\right)+i \sin \left(22.5^{\circ}\right)\right], 4\left[\cos \left(112.5^{\circ}\right)+i \sin \left(112.5^{\circ}\right)\right]\right. $$ $$ \left.4\left[\cos \left(247.5^{\circ}\right)+i \sin \left(247.5^{\circ}\right)\right], 4\left[\cos \left(337.5^{\circ}\right)+i \sin \left(337.5^{\circ}\right)\right]\right\} $$ D. $$ \left\{4\left[\cos \left(67.5^{\circ}\right)+i \sin \left(67.5^{\circ}\right)\right], 4\left[\cos \left(157.5^{\circ}\right)+i \sin \left(157.5^{\circ}\right)\right]\right. $$ $$ \left.4\left[\cos \left(202.5^{\circ}\right)+i \sin \left(202.5^{\circ}\right)\right], 4\left[\cos \left(292.5^{\circ}\right)+i \sin \left(292.5^{\circ}\right)\right]\right\} $$

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To solve the equation $$ x^4 + 256i = 0 $$, we can rewrite it as:

$$
x^4 = -256i
$$

Next, we need to express $$-256i$$ in polar form. The modulus of $$-256i$$ is:

$$
| -256i | = 256
$$

The argument (angle) of $$-256i$$ is:

$$
\text{arg}(-256i) = \frac{3\pi}{2} \quad \text{(or } 270^\circ\text{)}
$$

Thus, we can express $$-256i$$ in polar form as:

$$
-256i = 256 \left( \cos\left(\frac{3\pi}{2}\right) + i \sin\left(\frac{3\pi}{2}\right) \right)
$$

Now, we can find the fourth roots of $$x^4 = -256i$$. The modulus of the roots will be:

$$
|x| = \sqrt[4]{256} = 4
$$

The argument of the roots will be:

$$
\frac{\frac{3\pi}{2} + 2k\pi}{4} \quad \text{for } k = 0, 1, 2, 3
$$

Calculating the angles for each $$k$$:

1. For $$k = 0$$:
   $$
   \theta_0 = \frac{\frac{3\pi}{2}}{4} = \frac{3\pi}{8} = 67.5^\circ
   $$

2. For $$k = 1$$:
   $$
   \theta_1 = \frac{\frac{3\pi}{2} + 2\pi}{4} = \frac{\frac{3\pi}{2} + \frac{8\pi}{4}}{4} = \frac{\frac{11\pi}{4}}{4} = \frac{11\pi}{16} = 157.5^\circ
   $$

3. For $$k = 2$$:
   $$
   \theta_2 = \frac{\frac{3\pi}{2} + 4\pi}{4} = \frac{\frac{3\pi}{2} + \frac{16\pi}{4}}{4} = \frac{\frac{19\pi}{4}}{4} = \frac{19\pi}{16} = 247.5^\circ
   $$

4. For $$k = 3$$:
   $$
   \theta_3 = \frac{\frac{3\pi}{2} + 6\pi}{4} = \frac{\frac{3\pi}{2} + \frac{24\pi}{4}}{4} = \frac{\frac{27\pi}{4}}{4} = \frac{27\pi}{16} = 337.5^\circ
   $$

Thus, the four roots of the equation are:

$$
x_k = 4 \left( \cos\left(\theta_k\right) + i \sin\left(\theta_k\right) \right) \quad \text{for } k = 0, 1, 2, 3
$$

This gives us the solutions:

$$
\{ 4(\cos(67.5^\circ) + i \sin(67.5^\circ)), 4(\cos(157.5^\circ) + i \sin(157.5^\circ)), 4(\cos(247.5^\circ) + i \sin(247.5^\circ)), 4(\cos(337.5^\circ) + i \sin(337.5^\circ)) \}
$$

Looking at the provided options, the correct solution set is:

**B.** $$ \left\{4\left[\cos \left(67.5^{\circ}\right)+i \sin \left(67.5^{\circ}\right)\right], 4\left[\cos \left(157.5^{\circ}\right)+i \sin \left(157.5^{\circ}\right)\right], 4\left[\cos \left(247.5^{\circ}\right)+i \sin \left(247.5^{\circ}\right)\right], 4\left[\cos \left(337.5^{\circ}\right)+i \sin \left(337.5^{\circ}\right)\right]\right\} $$
The correct solution set is B.

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.