Find the cube roots of the following complex number. Then plot the cube roots. $$ -5-5 \sqrt{3} i $$ Choose the three cube roots of $$ -5-5 \sqrt{3} i $$. A. $$ \sqrt[3]{10} $$ cis $$ 110^{\circ}, \sqrt[3]{10} $$ cis $$ 230^{\circ}, \sqrt[3]{10} $$ cis $$ 350^{\circ} $$ B. $$ \sqrt[3]{10} $$ cis $$ 80^{\circ}, \sqrt[3]{10} $$ cis $$ 200^{\circ}, \sqrt[3]{10} $$ cis $$ 320^{\circ} $$ C. 10 cis $$ 80^{\circ}, 10 $$ cis $$ 200^{\circ}, 10 $$ cis $$ 320^{\circ} $$ D. 10 cis $$ 110^{\circ}, 10 $$ cis $$ 230^{\circ}, 10 $$ cis $$ 350^{\circ} $$ D.

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UpStudy AI · Accepted Answer

To find the cube roots of the complex number $$ -5-5 \sqrt{3} i $$, we can use De Moivre's Theorem.

The cube roots of a complex number $$ z $$ are given by:
$$ \sqrt[3]{z} = \sqrt[3]{|z|} \left( \cos\left(\frac{\arg(z) + 2k\pi}{3}\right) + i \sin\left(\frac{\arg(z) + 2k\pi}{3}\right) \right) $$
where $$ k = 0, 1, 2 $$ and $$ \arg(z) $$ is the argument of the complex number.

Given complex number: $$ z = -5-5 \sqrt{3} i $$

1. Find the magnitude of $$ z $$:
$$ |z| = \sqrt{(-5)^2 + (-5 \sqrt{3})^2} = \sqrt{25 + 75} = \sqrt{100} = 10 $$

2. Find the argument of $$ z $$:
$$ \arg(z) = \arctan\left(\frac{-5 \sqrt{3}}{-5}\right) = \arctan(\sqrt{3}) = 60^{\circ} $$

3. Calculate the cube roots:
$$ \sqrt[3]{z} = \sqrt[3]{10} \left( \cos\left(\frac{60^{\circ} + 2k\pi}{3}\right) + i \sin\left(\frac{60^{\circ} + 2k\pi}{3}\right) \right) $$
where $$ k = 0, 1, 2 $$

Let's calculate the cube roots and plot them.
Calculate the value by following steps:
- step0: Calculate:
  $$\sqrt[3]{10}$$
Calculate or simplify the expression $$ \cos\left(\frac{60^{\circ} + 2\pi}{3}\right) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\cos\left(\frac{60^{\circ}+2\pi }{3}\right)$$
- step1: Add the terms:
  $$\cos\left(\frac{\frac{7\pi }{3}}{3}\right)$$
- step2: Divide the terms:
  $$\cos\left(\frac{7\pi }{9}\right)$$
- step3: Rewrite the expression:
  $$-\cos\left(\frac{2\pi }{9}\right)$$
Calculate or simplify the expression $$ \sin\left(\frac{60^{\circ} + 2\pi}{3}\right) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\sin\left(\frac{60^{\circ}+2\pi }{3}\right)$$
- step1: Add the terms:
  $$\sin\left(\frac{\frac{7\pi }{3}}{3}\right)$$
- step2: Divide the terms:
  $$\sin\left(\frac{7\pi }{9}\right)$$
The cube roots of the complex number $$ -5-5 \sqrt{3} i $$ are:
1. $$ \sqrt[3]{10} $$ cis $$ 80^{\circ} $$
2. $$ \sqrt[3]{10} $$ cis $$ 200^{\circ} $$
3. $$ \sqrt[3]{10} $$ cis $$ 320^{\circ} $$

Therefore, the correct answer is B. $$ \sqrt[3]{10} $$ cis $$ 80^{\circ}, \sqrt[3]{10} $$ cis $$ 200^{\circ}, \sqrt[3]{10} $$ cis $$ 320^{\circ} $$.
The cube roots of $$ -5-5 \sqrt{3} i $$ are $$ \sqrt[3]{10} $$ cis $$ 80^{\circ}, \sqrt[3]{10} $$ cis $$ 200^{\circ}, \sqrt[3]{10} $$ cis $$ 320^{\circ} $$. Therefore, the correct answer is B.

Find the cube roots of the following complex number. Then plot the cube roots.

Choose the three cube roots of .
A. cis cis cis
B. cis cis cis
C. 10 cis cis cis
D. 10 cis cis cis
D.

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Find the cube roots of the following complex number. Then plot the cube roots. Choose the three cube roots of . A. cis cis cis B. cis cis cis C. 10 cis cis cis D. 10 cis cis cis D.