find all the roots to the equation: z^3 = 1 + i in polar form

1st write 1+ i in polar form, sketch a diagram to find the angle ( =pi/4) and find the modulus ( sqrt(2))z^3 = Sqrt(2) E^ipi/4This is true for all equivalent solutions (add 2kpi) z^3 = sqrt(2)e^(pi/4 +2kpi)iUse De moivres theorem: z = 2^(1/6) e^(pi/12 +2kpi/3)iThis is an algebraic equation, so has 3 solutions (since z^3 is the highest power) answers are usually given with angles in range -pi < x < pi. So our solutions correspond to k = 0, 1, -1z = 2^1/6 e^ipi/12 , 2^1/6 e^i3pi/4 , 2^1/6 e^-i7pi/12

RM
Answered by Rajan M. Further Mathematics tutor

9346 Views

See similar Further Mathematics A Level tutors

Related Further Mathematics A Level answers

All answers ▸

Find the displacement function if the acceleration function is a=2t+5. Assume a zero initial condition of displacement and v=8 when t=1.


Are the integers a group under addition? How about multiplication?


Find the square roots of 2 + isqrt(5)


Using the definitions of hyperbolic functions in terms of exponentials show that sech^2(x) = 1-tanh^2(x)


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences