Answer to Question #148914 in Algebra for Dhruv bartwal

Question

Find all the 5th roots of (5-1)

Answer 1

Let

a=5-i

IaI=$\sqrt{ {5}^2+({-1})^2}\\$

IaI=$\sqrt{26}\\$

$\alpha = tan^{-1}(-5)\\ a=|a|e^{i\alpha}=|a|(cos\alpha+isin\alpha)\\ a^{\frac{1}{5}}=|a|^{\frac{1}{5}}e^{i\frac{\alpha}{5}}\\ a^{\frac{1}{5}}=\sqrt{26}^{\frac{1}{5}}(cos\frac{tan^{-1}(\frac{1}{-5})+2kπ}{5}+isin\frac{tan^{-1}(\frac{1}{-5})+2kπ}{5})\\ k= 0, 1, 2, 3, 4$