How do you find the cube root of z = 1 + i?

Firstly we express z in polar form:


z = R*ei*θ


where |z| = (Re2 + Im2)0.5 = (12 + 12)0.5 = 20.5


θ = arg z = tan-1(Im/Re) = tan-1(1/1) = π/4


Therefore z = (20.5)*ei*π/4


We can add on any multiple of 2π to the argument of z without affecting the value of the complex number:


z = (20.5)*ei*(π/4 + 2*π*n)


where n is an integer


We then take cube roots of both sides (not forgetting to cube root the modulus R as well as the exponent):


z1/3 = (21/6)*ei(π/12 + 2*π*n/3) = (21/6)*ei(π + 8*π*n)/12


Because we are calculating the cube root, we expect three solutions. To find these three roots, we substitute in three consecutive integers into n. We will choose n = 0, 1, 2.


Solution 1 (with n=0): z1/3 = (21/6)*ei(π/12)

Solution 2 (with n=1): z1/3 = (21/6)*ei(3π/4)

Solution 3 (with n=2): z1/3 = (21/6)*ei(17π/12)


We can convert these back into Cartesian form using:


z = R*(cosθ + i sinθ)


We find that:


Solution 1: z1/3 =(21/6)*(cos(π/12) + i sin(π/12)) = 1.08 + 0.291i

Solution 2: z1/3 = (21/6)*(cos(3π/4) + i sin(3π/4)) = -0.794 +0.794i

Solution 3: z1/3 = (21/6)*(cos(17π/12) + i sin(17π/12)) = -0.291-1.084i

Aldo E. GCSE Maths tutor, A Level Maths tutor, A Level Further Mathem...

2 years ago

Answered by Aldo, an A Level Further Mathematics tutor with MyTutor

Still stuck? Get one-to-one help from a personally interviewed subject specialist


£20 /hr

Liberty A.

Degree: Mathematics (Bachelors) - Durham University

Subjects offered: Further Mathematics , Physics+ 1 more

Further Mathematics

“About Me Hi I'm Libby, I am in my first year studying Mathematics at Durham University and live in Buckinghamshire. I believe Mathematics and Pysics are fantastic subjects and I enjoy tutoring as I am able to share my passion and help...”

MyTutor guarantee

£22 /hr

Tadas T.

Degree: MMathPhil Mathematics and Philosophy (Bachelors) - Oxford, St Anne's College University

Subjects offered: Further Mathematics , Philosophy and Ethics+ 5 more

Further Mathematics
Philosophy and Ethics
-Personal Statements-
-Oxbridge Preparation-

“Currently I am a third year Maths and Philosophy student at the University of Oxford. I have been interested in both Maths and Philosophy for quite a long time now and I hope I can pass both the interest and knowledge for the subject(...”

£20 /hr

Leo P.

Degree: Theoretical Physics (Masters) - University College London University

Subjects offered: Further Mathematics , Maths

Further Mathematics

“About Me: I am a MSci Theoritical Physics student at University College London (UCL). Maths is everywhere and that's what I love about maths. Hopefully you'll enjoy it too I am a friendly, understanding and patient tutor that enjoys t...”

MyTutor guarantee

About the author

Aldo E.

Currently unavailable: for regular students

Degree: Engineering (Masters) - Cambridge University

Subjects offered: Further Mathematics , Physics+ 1 more

Further Mathematics

“Hi, I'm Aldo! I'm a Cambridge engineer and I'm passionate about maths and physics, as well as passing on knowledge to others!”

You may also like...

Posts by Aldo

How do you calculate the Earth's escape velocity?

How do you find the cube root of z = 1 + i?

Other A Level Further Mathematics questions

Using mathematical induction, prove that n^3+2n is divisible by 3 for all integers n

Express (2x-1)/(x-1)(2x-3) in partial fractions.

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.

Find the square root of i

View A Level Further Mathematics tutors


We use cookies to improve our service. By continuing to use this website, we'll assume that you're OK with this. Dismiss