Use de Moivres theorem to evaluate 34i7 Express the results

Use de Moivre\'s theorem to evaluate (3+4i)^7. Express the results in trigonometric form

Solution

(3+ 4i)⁷

r =(3²+4²)=5

=(5((3/5)+ (4/5)i))⁷

=5⁷((3/5)+ (4/5)i)⁷

=5⁷(cos(cos^-1(3/5))+ sin(cos^-1(3/5))i)⁷

=5⁷(cos(7cos^-1(3/5))+ sin(7cos^-1(3/5))i)