Roots of a Complex Number Calculator

Find the roots of a complex number, roots of unity step by step

The calculator will find the nn-th roots of the given complex number using de Moivre's formula, with steps shown.

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Your Input

Find 81i4\sqrt[4]{81 i}.

Solution

The polar form of 81i81 i is 81(cos(π2)+isin(π2))81 \left(\cos{\left(\frac{\pi}{2} \right)} + i \sin{\left(\frac{\pi}{2} \right)}\right) (for steps, see polar form calculator).

According to the De Moivre's Formula, all nn-th roots of a complex number r(cos(θ)+isin(θ))r \left(\cos{\left(\theta \right)} + i \sin{\left(\theta \right)}\right) are given by r1n(cos(θ+2πkn)+isin(θ+2πkn))r^{\frac{1}{n}} \left(\cos{\left(\frac{\theta + 2 \pi k}{n} \right)} + i \sin{\left(\frac{\theta + 2 \pi k}{n} \right)}\right), k=0..n1k=\overline{0..n-1}.

We have that r=81r = 81, θ=π2\theta = \frac{\pi}{2}, and n=4n = 4.

  • k=0k = 0: 814(cos(π2+2π04)+isin(π2+2π04))=3(cos(π8)+isin(π8))=324+12+3i1224\sqrt[4]{81} \left(\cos{\left(\frac{\frac{\pi}{2} + 2\cdot \pi\cdot 0}{4} \right)} + i \sin{\left(\frac{\frac{\pi}{2} + 2\cdot \pi\cdot 0}{4} \right)}\right) = 3 \left(\cos{\left(\frac{\pi}{8} \right)} + i \sin{\left(\frac{\pi}{8} \right)}\right) = 3 \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + 3 i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}
  • k=1k = 1: 814(cos(π2+2π14)+isin(π2+2π14))=3(cos(5π8)+isin(5π8))=31224+3i24+12\sqrt[4]{81} \left(\cos{\left(\frac{\frac{\pi}{2} + 2\cdot \pi\cdot 1}{4} \right)} + i \sin{\left(\frac{\frac{\pi}{2} + 2\cdot \pi\cdot 1}{4} \right)}\right) = 3 \left(\cos{\left(\frac{5 \pi}{8} \right)} + i \sin{\left(\frac{5 \pi}{8} \right)}\right) = - 3 \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + 3 i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}
  • k=2k = 2: 814(cos(π2+2π24)+isin(π2+2π24))=3(cos(9π8)+isin(9π8))=324+123i1224\sqrt[4]{81} \left(\cos{\left(\frac{\frac{\pi}{2} + 2\cdot \pi\cdot 2}{4} \right)} + i \sin{\left(\frac{\frac{\pi}{2} + 2\cdot \pi\cdot 2}{4} \right)}\right) = 3 \left(\cos{\left(\frac{9 \pi}{8} \right)} + i \sin{\left(\frac{9 \pi}{8} \right)}\right) = - 3 \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - 3 i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}
  • k=3k = 3: 814(cos(π2+2π34)+isin(π2+2π34))=3(cos(13π8)+isin(13π8))=312243i24+12\sqrt[4]{81} \left(\cos{\left(\frac{\frac{\pi}{2} + 2\cdot \pi\cdot 3}{4} \right)} + i \sin{\left(\frac{\frac{\pi}{2} + 2\cdot \pi\cdot 3}{4} \right)}\right) = 3 \left(\cos{\left(\frac{13 \pi}{8} \right)} + i \sin{\left(\frac{13 \pi}{8} \right)}\right) = 3 \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - 3 i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}

Answer

81i4=324+12+3i12242.77163859753386+1.148050297095269i\sqrt[4]{81 i} = 3 \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + 3 i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\approx 2.77163859753386 + 1.148050297095269 iA

81i4=31224+3i24+121.148050297095269+2.77163859753386i\sqrt[4]{81 i} = - 3 \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + 3 i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\approx -1.148050297095269 + 2.77163859753386 iA

81i4=324+123i12242.771638597533861.148050297095269i\sqrt[4]{81 i} = - 3 \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - 3 i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\approx -2.77163859753386 - 1.148050297095269 iA

81i4=312243i24+121.1480502970952692.77163859753386i\sqrt[4]{81 i} = 3 \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - 3 i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\approx 1.148050297095269 - 2.77163859753386 iA