Consider the equation z1313i Find the value of z which satis

Consider the equation z^13=(1+3i). Find the value of z which satisfies this equation and which has the second smallest positive argument ,0<<2 Express your answer as z=r^e where r=? and theta=?

Solution

z^13 = (1 +sqrt(3)*i)

r = sqrt( 1^2 + (sqrt3)^2 ) = 2

= tan^-1(sqrt3) = pi/3

z = (1 +sqrt(3)*i)^1/13 = (re^i)^1/13

= (2e^ipi/3)^1/13

z = 2^1/13 e^(i*pi/39)

= 2^1/3(0.9967 + i*0.08)

= 1.05 + i*0.084