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You asked:
Investigate the equation:
z 4 = 1 − i {z}^{4} = 1 - i z 4 = 1 − i .
MathBot Answer:
The 4 solutions to the equation are:
z = 2 8 cos ( 7 π 16 ) + 2 8 i sin ( 7 π 16 ) ≈ 0.2127475 + 1.0695539 i z = − 2 8 cos ( π 16 ) + 2 8 i sin ( π 16 ) ≈ − 1.0695539 + 0.2127475 i z = 2 8 cos ( π 16 ) − 2 8 i sin ( π 16 ) ≈ 1.0695539 − 0.2127475 i z = − 2 8 cos ( 7 π 16 ) − 2 8 i sin ( 7 π 16 ) ≈ − 0.2127475 − 1.0695539 i \begin{aligned}z &= \sqrt[8]{2} \cos{\left(\frac{7 \pi}{16} \right)} + \sqrt[8]{2} i \sin{\left(\frac{7 \pi}{16} \right)} \approx 0.2127475 + 1.0695539 i\\z &= - \sqrt[8]{2} \cos{\left(\frac{\pi}{16} \right)} + \sqrt[8]{2} i \sin{\left(\frac{\pi}{16} \right)} \approx -1.0695539 + 0.2127475 i\\z &= \sqrt[8]{2} \cos{\left(\frac{\pi}{16} \right)} - \sqrt[8]{2} i \sin{\left(\frac{\pi}{16} \right)} \approx 1.0695539 -0.2127475 i\\z &= - \sqrt[8]{2} \cos{\left(\frac{7 \pi}{16} \right)} - \sqrt[8]{2} i \sin{\left(\frac{7 \pi}{16} \right)} \approx -0.2127475 -1.0695539 i\end{aligned} z z z z = 8 2 cos ( 16 7 π ) + 8 2 i sin ( 16 7 π ) ≈ 0.2127475 + 1.0695539 i = − 8 2 cos ( 16 π ) + 8 2 i sin ( 16 π ) ≈ − 1.0695539 + 0.2127475 i = 8 2 cos ( 16 π ) − 8 2 i sin ( 16 π ) ≈ 1.0695539 − 0.2127475 i = − 8 2 cos ( 16 7 π ) − 8 2 i sin ( 16 7 π ) ≈ − 0.2127475 − 1.0695539 i
i i i imaginary unit , defined as i 2 = − 1 i^2 = -1 i 2 = − 1
