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Question 1 of 5Simplify, write in rectangular form.
$$\left[3\left(\text{cos}\frac{5π}{6}+ i \hspace{.1em}\text{sin}\hspace{.1em}\frac{5π}{6}\right)\right]^{4}$$
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Select the Correct Answer Below: Correct! Cool Emoji Not Correct! Cool Emoji
A
$$-338 - 338i\sqrt{3}$$
B
$$-\frac{81}{2}- \frac{81\sqrt{3}}{2}i$$
C
$$-392 + 392i\sqrt{3}$$
D
$$-128 + 128i\sqrt{3}$$
E
$$-\frac{97}{2}+ \frac{13\sqrt{5}}{2}i$$
Question 2 of 5Simplify, write in rectangular form.
$$(5 - 2i)^{5}$$
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A
$$-1475 - 4282i$$
B
$$-16 + 16i\sqrt{3}$$
C
$$972 - 972i$$
D
$$\frac{243\sqrt{3}}{2}+ \frac{243}{2}i$$
E
$$\frac{219\sqrt{7}}{2}- \frac{549}{2}i$$
Question 3 of 5Find all nth roots in polar form.
$$-\sqrt{3}- i, n=5$$
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A
$$\sqrt[5]{2}\left(\text{cos}\frac{π}{10}+ i \hspace{.1em}\text{sin}\hspace{.1em}\frac{π}{10}\right)$$ $$\sqrt[5]{2}\left(\text{cos}\frac{π}{2}+ i \hspace{.1em}\text{sin}\hspace{.1em}\frac{π}{2}\right)$$ $$\sqrt[5]{2}\left(\text{cos}\frac{9π}{10}+ i \hspace{.1em}\text{sin}\hspace{.1em}\frac{9π}{10}\right)$$ $$\sqrt[5]{2}\left(\text{cos}\frac{13π}{10}+ i \hspace{.1em}\text{sin}\hspace{.1em}\frac{13π}{10}\right)$$ $$\sqrt[5]{2}\left(\text{cos}\frac{17π}{10}+ i \hspace{.1em}\text{sin}\hspace{.1em}\frac{17π}{10}\right)$$
B
$$\sqrt[5]{2}\left(\text{cos}\frac{π}{3}+ i \hspace{.1em}\text{sin}\hspace{.1em}\frac{π}{3}\right)$$ $$\sqrt[5]{2}\left(\text{cos}\frac{11π}{15}+ i \hspace{.1em}\text{sin}\hspace{.1em}\frac{11π}{15}\right)$$ $$\sqrt[5]{2}\left(\text{cos}\frac{17π}{15}+ i \hspace{.1em}\text{sin}\hspace{.1em}\frac{17π}{15}\right)$$ $$\sqrt[5]{2}\left(\text{cos}\frac{23π}{15}+ i \hspace{.1em}\text{sin}\hspace{.1em}\frac{23π}{15}\right)$$ $$\sqrt[5]{2}\left(\text{cos}\frac{29π}{15}+ i \hspace{.1em}\text{sin}\hspace{.1em}\frac{29π}{15}\right)$$
C
$$\sqrt[5]{2}\left(\text{cos}\frac{2π}{15}+ i \hspace{.1em}\text{sin}\hspace{.1em}\frac{2π}{15}\right)$$ $$\sqrt[5]{2}\left(\text{cos}\frac{8π}{15}+ i \hspace{.1em}\text{sin}\hspace{.1em}\frac{8π}{15}\right)$$ $$\sqrt[5]{2}\left(\text{cos}\frac{14π}{15}+ i \hspace{.1em}\text{sin}\hspace{.1em}\frac{14π}{15}\right)$$ $$\sqrt[5]{2}\left(\text{cos}\frac{4π}{3}+ i \hspace{.1em}\text{sin}\hspace{.1em}\frac{4π}{3}\right)$$ $$\sqrt[5]{2}\left(\text{cos}\frac{26π}{15}+ i \hspace{.1em}\text{sin}\hspace{.1em}\frac{26π}{15}\right)$$
D
$$\sqrt[5]{2}\left(\text{cos}\frac{7π}{15}+ i \hspace{.1em}\text{sin}\hspace{.1em}\frac{7π}{15}\right)$$ $$\sqrt[5]{2}\left(\text{cos}\frac{13π}{15}+ i \hspace{.1em}\text{sin}\hspace{.1em}\frac{13π}{15}\right)$$ $$\sqrt[5]{2}\left(\text{cos}\frac{19π}{15}+ i \hspace{.1em}\text{sin}\hspace{.1em}\frac{19π}{15}\right)$$ $$\sqrt[5]{2}\left(\text{cos}\frac{5π}{3}+ i \hspace{.1em}\text{sin}\hspace{.1em}\frac{5π}{3}\right)$$ $$\sqrt[5]{2}\left(\text{cos}\frac{37π}{15}+ i \hspace{.1em}\text{sin}\hspace{.1em}\frac{37π}{15}\right)$$
E
$$\sqrt[5]{2}\left(\text{cos}\frac{7π}{30}+ i \hspace{.1em}\text{sin}\hspace{.1em}\frac{7π}{30}\right)$$ $$\sqrt[5]{2}\left(\text{cos}\frac{19π}{30}+ i \hspace{.1em}\text{sin}\hspace{.1em}\frac{19π}{30}\right)$$ $$\sqrt[5]{2}\left(\text{cos}\frac{31π}{30}+ i \hspace{.1em}\text{sin}\hspace{.1em}\frac{31π}{30}\right)$$ $$\sqrt[5]{2}\left(\text{cos}\frac{43π}{30}+ i \hspace{.1em}\text{sin}\hspace{.1em}\frac{43π}{30}\right)$$ $$\sqrt[5]{2}\left(\text{cos}\frac{11π}{6}+ i \hspace{.1em}\text{sin}\hspace{.1em}\frac{11π}{6}\right)$$
Question 4 of 5Find all nth roots in polar form.
$$-\frac{\sqrt{46}}{2}+ \frac{\sqrt{46}}{2}i, n=4$$
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A
$$\sqrt[8]{23}\left(\text{cos}\hspace{.1em}52.5° + i \hspace{.1em}\text{sin}\hspace{.1em}52.5°\right)$$ $$\sqrt[8]{23}\left(\text{cos}\hspace{.1em}142.5° + i \hspace{.1em}\text{sin}\hspace{.1em}142.5°\right)$$ $$\sqrt[8]{23}\left(\text{cos}\hspace{.1em}232.5° + i \hspace{.1em}\text{sin}\hspace{.1em}232.5°\right)$$ $$\sqrt[8]{23}\left(\text{cos}\hspace{.1em}322.5° + i \hspace{.1em}\text{sin}\hspace{.1em}322.5°\right)$$
B
$$\sqrt[4]{5}\left(\text{cos}\hspace{.1em}0° + i \hspace{.1em}\text{sin}\hspace{.1em}0°\right)$$ $$\sqrt[4]{5}\left(\text{cos}\hspace{.1em}90° + i \hspace{.1em}\text{sin}\hspace{.1em}90°\right)$$ $$\sqrt[4]{5}\left(\text{cos}\hspace{.1em}180° + i \hspace{.1em}\text{sin}\hspace{.1em}180°\right)$$ $$\sqrt[4]{5}\left(\text{cos}\hspace{.1em}270° + i \hspace{.1em}\text{sin}\hspace{.1em}270°\right)$$
C
$$\sqrt[4]{5}\left(\text{cos}\hspace{.1em}30° + i \hspace{.1em}\text{sin}\hspace{.1em}30°\right)$$ $$\sqrt[4]{5}\left(\text{cos}\hspace{.1em}120° + i \hspace{.1em}\text{sin}\hspace{.1em}120°\right)$$ $$\sqrt[4]{5}\left(\text{cos}\hspace{.1em}210° + i \hspace{.1em}\text{sin}\hspace{.1em}210°\right)$$ $$\sqrt[4]{5}\left(\text{cos}\hspace{.1em}300° + i \hspace{.1em}\text{sin}\hspace{.1em}300°\right)$$
D
$$\sqrt[4]{23}\left(\text{cos}\hspace{.1em}33.75° + i \hspace{.1em}\text{sin}\hspace{.1em}33.75°\right)$$ $$\sqrt[4]{23}\left(\text{cos}\hspace{.1em}123.75° + i \hspace{.1em}\text{sin}\hspace{.1em}123.75°\right)$$ $$\sqrt[4]{23}\left(\text{cos}\hspace{.1em}213.75° + i \hspace{.1em}\text{sin}\hspace{.1em}213.75°\right)$$ $$\sqrt[4]{23}\left(\text{cos}\hspace{.1em}303.75° + i \hspace{.1em}\text{sin}\hspace{.1em}303.75°\right)$$
E
$$\sqrt[8]{23}\left(\text{cos}\hspace{.1em}33.75° + i \hspace{.1em}\text{sin}\hspace{.1em}33.75°\right)$$ $$\sqrt[8]{23}\left(\text{cos}\hspace{.1em}123.75° + i \hspace{.1em}\text{sin}\hspace{.1em}123.75°\right)$$ $$\sqrt[8]{23}\left(\text{cos}\hspace{.1em}213.75° + i \hspace{.1em}\text{sin}\hspace{.1em}213.75°\right)$$ $$\sqrt[8]{23}\left(\text{cos}\hspace{.1em}303.75° + i \hspace{.1em}\text{sin}\hspace{.1em}303.75°\right)$$
Question 5 of 5Find all complex solutions in polar form.
$$x^4 + 1=0$$
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A
$$\text{cos}\hspace{.1em}25° + i \hspace{.1em}\text{sin}\hspace{.1em}25°$$ $$\text{cos}\hspace{.1em}135° + i \hspace{.1em}\text{sin}\hspace{.1em}135°$$ $$\text{cos}\hspace{.1em}245° + i \hspace{.1em}\text{sin}\hspace{.1em}245°$$ $$\text{cos}\hspace{.1em}355° + i \hspace{.1em}\text{sin}\hspace{.1em}355°$$
B
$$\text{cos}\hspace{.1em}95° + i \hspace{.1em}\text{sin}\hspace{.1em}95°$$ $$\text{cos}\hspace{.1em}165° + i \hspace{.1em}\text{sin}\hspace{.1em}165°$$ $$\text{cos}\hspace{.1em}235° + i \hspace{.1em}\text{sin}\hspace{.1em}235°$$ $$\text{cos}\hspace{.1em}305° + i \hspace{.1em}\text{sin}\hspace{.1em}305°$$
C
$$\text{cos}\hspace{.1em}45° + i \hspace{.1em}\text{sin}\hspace{.1em}45°$$ $$\text{cos}\hspace{.1em}135° + i \hspace{.1em}\text{sin}\hspace{.1em}135°$$ $$\text{cos}\hspace{.1em}225° + i \hspace{.1em}\text{sin}\hspace{.1em}225°$$ $$\text{cos}\hspace{.1em}315° + i \hspace{.1em}\text{sin}\hspace{.1em}315°$$
D
$$\sqrt[4]{2}(\text{cos}\hspace{.1em}45° + i \hspace{.1em}\text{sin}\hspace{.1em}45°)$$ $$\sqrt[4]{2}(\text{cos}\hspace{.1em}135° + i \hspace{.1em}\text{sin}\hspace{.1em}135°)$$ $$\sqrt[4]{2}(\text{cos}\hspace{.1em}225° + i \hspace{.1em}\text{sin}\hspace{.1em}225°)$$ $$\sqrt[4]{2}(\text{cos}\hspace{.1em}315° + i \hspace{.1em}\text{sin}\hspace{.1em}315°)$$
E
$$4(\text{cos}\hspace{.1em}95° + i \hspace{.1em}\text{sin}\hspace{.1em}95°)$$ $$4(\text{cos}\hspace{.1em}165° + i \hspace{.1em}\text{sin}\hspace{.1em}165°)$$ $$4(\text{cos}\hspace{.1em}235° + i \hspace{.1em}\text{sin}\hspace{.1em}235°)$$ $$4(\text{cos}\hspace{.1em}305° + i \hspace{.1em}\text{sin}\hspace{.1em}305°)$$

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