Find the fourth roots of 16(cos 200° + i sin 200°).

Solution:

Given 16(cos 200° + i sin 200°).

Let y = 16.eⁱ²⁰⁰

Apply square root to get desired value

(2⁴)^1/2. (eⁱ²⁰⁰)^1/2

= 4(eⁱ¹⁰⁰)

Again apply square root to get four roots

(2²)^1/2 . (eⁱ¹⁰⁰)^1/2

2.eⁱ⁵⁰

y = 2(cos(50 + 90n) + i sin(50 + 90n))

Take n = 0 ,y = 2(cos(50)+isin(50)) : 2(0.64+i0.766)

Take n = 1,y = 2(cos(140) + isin (140)): 2(-0.766 + i0.64)

Take n = 2,y = 2(cos(230) + isin (230)): 2(-0.64 + i(-0.766))

Take n = 3,y = 2(cos(320) + isin(320)): 2(0.766 + i(-0.64))

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Find the fourth roots of 16(cos 200° + i sin 200°).

Summary:

The fourth roots of 16(cos 200° + i sin 200°) are 2(0.64 + i0.766), 2(-0.766 + i0.64), 2(-0.64 + i(-0.766)), 2(0.766 + i(-0.64)).