Answer:
0.7239 +0.9435i . . . and 3 others
Step-by-step explanation:
Roots of complex numbers are most easily found when the numbers are in polar form. Then Euler's formula can be used.
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polar/exponential form
The given number, written in polar and/or exponential form is ...
z⁴ = -√3 -i = 2∠210° = 2·e^(i(7π/6+2kπ))
roots
The value of z is the fourth root of this. The fourth root of 2 multiplies the exponential term with 1/4 of the original exponent. We note there are four roots.
[tex]\displaystyle z=\sqrt[4]{2\cdot e^{i(7\pi/6+2k\pi)}}=\sqrt[4]{2}\cdot e^{i(7\pi/24+k\pi/2)}\\\\=\begin{cases}(\sqrt[4]{2}\cos(52.5^\circ)+i\sqrt[4]{2}\sin(52.5^\circ))&\approx 0.7239+0.9435i\\(\sqrt[4]{2}\cos(142.5^\circ)+i\sqrt[4]{2}\sin(142.5^\circ))&\approx-0.9435+0.7239i\\(\sqrt[4]{2}\cos(232.5^\circ)+i\sqrt[4]{2}\sin(232.5^\circ))&\approx -0.7239-0.9435i\\(\sqrt[4]{2}\cos(322.5^\circ)+i\sqrt[4]{2}\sin(322.5^\circ))&\approx 0.9435-0.7239i\end{cases}[/tex]
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Additional comment
The value of z⁴ can also be written as 2∠-150°. The root found by the calculator (second attachment) has 1/4 of this negative angle, rather than 1/4 of the angle +210°.
