How do you find the 4rd root of #81e^(60i)#? Trigonometry The Polar System De Moivre’s and the nth Root Theorems 1 Answer Salvatore I. Nov 10, 2016 #3e^(i(15+kpi/2))# Four roots each for k=0,1,2,3 Explanation: #z^4=81e^(i(60+2kpi))# #z=root4(81)e^(i(60+2kpi)/4)=3e^(i(15+kpi/2))# Answer link Related questions How do you use De Moivre’s Theorem to find the powers of complex numbers in polar form? What is the DeMoivre's theorem used for? How do you find the #n^{th}# roots of complex numbers in polar form? How do you find #[2(\cos 120^\circ + i \sin 120^\circ)]^5# using the De Moivre's theorem? What is #(-\frac{1}{2}+\frac{i\sqrt{3}}{2})^{10}#? How do you find the three cube roots of #-2-2i \sqrt{3}#? If the roots can be determined, will some form of De Moivre’s Theorem be used? How do you find the two square roots of 2i? Question #f6317 How do you find #z, z^2, z^3, z^4# given #z=sqrt2/2(1+i)#? See all questions in De Moivre’s and the nth Root Theorems Impact of this question 1776 views around the world You can reuse this answer Creative Commons License