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Nataliya [291]
2 years ago
9

ss="latex-formula">

z = ____ + ____ i
Mathematics
1 answer:
abruzzese [7]2 years ago
3 0

If <em>z</em> ⁷ = 128<em>i</em>, then there are 7 complex numbers <em>z</em> that satisfy this equation.

z^7 = 128i = 2^7i = 2^7e^{i\frac\pi2}

\implies z=\sqrt[7]{2^7} e^{i\frac17\left(\frac\pi2+2n\pi\right)}

(where <em>n</em> = 0, 1, 2, …, 6)

\implies z = 2 e^{i\left(\frac\pi{14}+\frac{2n\pi}7\right)}

\displaystyle\implies z = 2 \left(\cos\left(\frac\pi{14}+\frac{2n\pi}7\right)+i\sin\left(\frac\pi{14}+\frac{2n\pi}7\right)\right)

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