Question

One of the roots of the equation \((x + 1)^4 + 81 = 0\) is

• A. \(3\left(\frac{1+i}{\sqrt{2}}\right)\)
• B. \(-\left(\frac{3+\sqrt{2}+3i}{\sqrt{2}}\right)\)
• C. \(-\left(\frac{3+\sqrt{2}+i}{\sqrt{2}}\right)\)
• D. \(-\left(\frac{3+3i}{\sqrt{2}}\right)\)

Hint:

For equations of the form \(z^n = -a\), the roots will always lie symmetrically on a circle of radius \(a^{1/n}\) in the complex plane. Checking angles corresponding to \(k = 0, 1, 2, \dots\) sequentially is the most reliable way to match with options.

Answer 1

The Correct Option is B