Obtaining roots of a complex number
Obtaining roots of a complex number
IB Year 6 | Grade 12
Solve the equations
- \(z^4 = 1 - i\)
- \(z^5 = 1 + \sqrt{3}i\)
- \((z + 2i)^5 = 1\)
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Answer:(a) \(z=2^{1/8}e^{-\pi i/16},\ 2^{1/8}e^{7\pi i/16},\ 2^{1/8}e^{15\pi i/16},\ 2^{1/8}e^{-9\pi i/16}\) (b) \(z=2^{1/5}e^{\pi i/15},\ 2^{1/5}e^{7\pi i/15},\ 2^{1/5}e^{13\pi i/15},\ 2^{1/5}e^{-11\pi i/15},\ 2^{1/5}e^{-\pi i/3}\) (c) \(z=e^{2k\pi i/5}-2i,\ k=0,1,2,3,4\)
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