asked by guest
You asked:
Investigate the equation:
z 5 = i {z}^{5} = i z 5 = i .
MathBot Answer:
The 5 solutions to the equation are:
z = i z = 5 8 − 5 8 + i ( − 5 4 − 1 4 ) ≈ 0.58778525 − 0.80901699 i z = 5 8 + 5 8 + i ( − 1 4 + 5 4 ) ≈ 0.95105652 + 0.30901699 i z = − 5 8 − 5 8 + i ( − 5 4 − 1 4 ) ≈ − 0.58778525 − 0.80901699 i z = − 5 8 + 5 8 + i ( − 1 4 + 5 4 ) ≈ − 0.95105652 + 0.30901699 i \begin{aligned}z &= i\\z &= \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} + i \left(- \frac{\sqrt{5}}{4} - \frac{1}{4}\right) \approx 0.58778525 -0.80901699 i\\z &= \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + i \left(- \frac{1}{4} + \frac{\sqrt{5}}{4}\right) \approx 0.95105652 + 0.30901699 i\\z &= - \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} + i \left(- \frac{\sqrt{5}}{4} - \frac{1}{4}\right) \approx -0.58778525 -0.80901699 i\\z &= - \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + i \left(- \frac{1}{4} + \frac{\sqrt{5}}{4}\right) \approx -0.95105652 + 0.30901699 i\end{aligned} z z z z z = i = 8 5 − 8 5 + i ( − 4 5 − 4 1 ) ≈ 0.58778525 − 0.80901699 i = 8 5 + 8 5 + i ( − 4 1 + 4 5 ) ≈ 0.95105652 + 0.30901699 i = − 8 5 − 8 5 + i ( − 4 5 − 4 1 ) ≈ − 0.58778525 − 0.80901699 i = − 8 5 + 8 5 + i ( − 4 1 + 4 5 ) ≈ − 0.95105652 + 0.30901699 i
i i i imaginary unit , defined as i 2 = − 1 i^2 = -1 i 2 = − 1
