TS EAMCET · Maths · Complex Number
One of the roots of the equation \((x+1)^4+81=0\) is
- A \(3\left(\frac{1+i}{\sqrt{2}}\right)\)
- B \(-\left(\frac{3+\sqrt{2}+3 i}{\sqrt{2}}\right)\)
- C \(-\left(\frac{3+\sqrt{2}+i}{\sqrt{2}}\right)\)
- D \(-\left(\frac{3+3 i}{\sqrt{2}}\right)\)
Answer & Solution
Correct Answer
(B) \(-\left(\frac{3+\sqrt{2}+3 i}{\sqrt{2}}\right)\)
Step-by-step Solution
Detailed explanation
\((x+1)^4 = -81\) \(x+1 = (-81)^{1/4}\) \(x+1 = 3 e^{i\frac{\pi+2k\pi}{4}}\) For \(k=2\): \(x+1 = 3 e^{i\frac{5\pi}{4}}\) \(x+1 = 3 \left(\cos\left(\frac{5\pi}{4}\right) + i\sin\left(\frac{5\pi}{4}\right)\right)\)…
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