z4=−1
z=4−1
The roots of a complex number are also given by a formula nz=n∣z∣(cosnφ+2πk+isinnφ+2πk), where k=0,n−1.
∣−1∣=1, φ=π
k=0:
z1=4−1=41(cos4π+2π⋅0+isin4π+2π⋅0)=1⋅(cos4π+isin4π)=22+i22
k=1:
z2=4−1=41(cos4π+2π⋅1+isin4π+2π⋅1)=1⋅(cos43π+isin43π)=22−i22
k=2:
z3=4−1=41(cos4π+2π⋅2+isin4π+2π⋅2)=1⋅(cos45π+isin45π)=−22−i22
k=3:
z4=4−1=41(cos4π+2π⋅3+isin4π+2π⋅3)=1⋅(cos47π+isin47π)=−22+i22
So z1=22+i22, z2=22−i22, z3=−22−i22, z4=−22+i22
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