Answer:
z = {1 +i, 1 -i, -1 +i, -1 -i}
Step-by-step explanation:
Using Euler's formula, you can write this as ...
z^4 = 4e^(i(π+2kπ))
Then the 4th root is ...
z = 4^(1/4)e^(i(π/4 +kπ/2)) = √2(cos(π/4+kπ/2) +i·sin(π/4+kπ/2))
for integers k in the range 0 to 3.
z = √2(±1/√2 ±i·1/√2) . . . signs are independent
z = ±1 ±i . . . . . . signs are independent; hence 4 solutions
z = {1 +i, 1 -i, -1 +i, -1 -i}
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Comment on roots of complex numbers
In the complex domain, each number has a number of roots equal to the index of the root. That is, there are four 4th roots. They all have the same magnitude, but their angles are separated by 360°/(root index) = 90°. Here, the principal fourth root of 4∠180° is 4^(1/4)∠(180°/4) = √2∠45° = 1+i.
Adding 90° is equivalent to multiplying by i, so the other 3 roots are √2∠135° = -1+i, √2∠225° = -1 -i, and √2∠315° = 1 -i.
