(5+7 i)^(1/3)
Assuming i is the imaginary unit | Use i as instead

Input

(5 + 7 i)^(1/3)

Decimal approximation

1.94698974704815246567826077141989372933418196671354349213990023681446153... + 0.638410039627964340187916808959351994912596197402251317636455358806146830... i

Alternate complex forms

1.9470 + 0.6384 i

74^(1/6) (cos(1/3 tan^(-1)(7/5)) + i sin(1/3 tan^(-1)(7/5)))

74^(1/6) e^(1/3 i tan^(-1)(7/5))

Position in the complex plane

Position in the complex plane

Polar coordinates

r = 74^(1/6) (radius), θ = 1/3 tan^(-1)(7/5) (angle)

Alternate form

74^(1/6) cos(1/3 tan^(-1)(7/5)) + i 74^(1/6) sin(1/3 tan^(-1)(7/5))

Minimal polynomial

x^6 - 10 x^3 + 74
The minimal polynomial of α is the irreducible polynomial with rational coefficients of smallest degree with α as a root »

Continued fraction

[2 + i; 3i, -2 + i, 2 - i, 3 + 5i, -4 - i, 1 + i, -2i, 2 + i, -3 - 2i, 2 - , ...] (using the Hurwitz expansion)

All 3rd roots of 5+7 i

74^(1/6) e^(1/3 i tan^(-1)(7/5)) ≈ 1.94699 + 0.6384 i (principal root)
74^(1/6) e^(1/3 i (2 π + tan^(-1)(7/5))) ≈ -1.5264 + 1.3669 i
74^(1/6) e^(i (-2 π + 1/3 (4 π + tan^(-1)(7/5)))) ≈ -0.4206 -2.0053 i

Plot of all roots in the complex plane

Plot of all roots in the complex plane
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