Question

Which of the following is a solution of x^4=-81

Answer 1

(c) 3(cos(135°) +i·sin(135°))

Explanation:

You want to identify a solution to x^4 = -81.

Roots

The principal fourth root of 81∠180° will be ...

`\sqrt[4]{-81}=\sqrt[4]{81\angle 180^\circ}=\sqrt[4]{81}\angle (180^\circ)/(4)=3\angle 45^\circ`

Additional complex roots will be at angles that are multiples of 360°/4 from this one, so ...

3∠135°, 3∠225°, 3∠315°

The root that is on the list of choices is ...

3∠135° = 3(cos(135°) +i·sin(135°))

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Additional comment

The attachment shows the 4th powers of the numbers with the various angles. The fourth power of 3∠135° gives -81.