The calculator will find the n-th roots of the given complex number using de Moivre's formula, with steps shown.
Solution
The polar form of 81i is 81(cos(2π)+isin(2π)) (for steps, see polar form calculator).
According to the De Moivre's Formula, all n-th roots of a complex number r(cos(θ)+isin(θ)) are given by rn1(cos(nθ+2πk)+isin(nθ+2πk)), k=0..n−1.
We have that r=81, θ=2π, and n=4.
- k=0: 481(cos(42π+2⋅π⋅0)+isin(42π+2⋅π⋅0))=3(cos(8π)+isin(8π))=342+21+3i21−42
- k=1: 481(cos(42π+2⋅π⋅1)+isin(42π+2⋅π⋅1))=3(cos(85π)+isin(85π))=−321−42+3i42+21
- k=2: 481(cos(42π+2⋅π⋅2)+isin(42π+2⋅π⋅2))=3(cos(89π)+isin(89π))=−342+21−3i21−42
- k=3: 481(cos(42π+2⋅π⋅3)+isin(42π+2⋅π⋅3))=3(cos(813π)+isin(813π))=321−42−3i42+21
Answer
481i=342+21+3i21−42≈2.77163859753386+1.148050297095269iA
481i=−321−42+3i42+21≈−1.148050297095269+2.77163859753386iA
481i=−342+21−3i21−42≈−2.77163859753386−1.148050297095269iA
481i=321−42−3i42+21≈1.148050297095269−2.77163859753386iA