MYTUTOR SUBJECT ANSWERS

684 views

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

83 SUBJECT SPECIALISTS

PremiumChristopher S. A Level Maths tutor, GCSE Maths tutor, A Level Spanish...
£36 /hr

Chris S.

Degree: Mathematics (Bachelors) - Bristol University

Subjects offered:Further Mathematics , Spanish+ 1 more

Further Mathematics
Spanish
Maths

“Top tutor from the renowned Russell university group, ready to help you improve your grades.”

£20 /hr

Izzy A.

Degree: Mathematics and Philosophy (Bachelors) - Durham University

Subjects offered:Further Mathematics , Maths

Further Mathematics
Maths

“I am a 19-year-old, currently in my 2nd year at Durham University studying Maths and Philosophy. ”

£26 /hr

Lloyd S.

Degree: Mathematics G100 (Bachelors) - Bristol University

Subjects offered:Further Mathematics , Maths

Further Mathematics
Maths

“About Me:I am 19 and from Devon, currently in my first year studying Maths at the University of Bristol. I have a real passion for Maths and I really hope I can help you to understand, and maybe even enjoy doing maths - I know not ev...”

About the author

Aldo E.

Currently unavailable: for regular students

Degree: Engineering (Masters) - Cambridge University

Subjects offered:Further Mathematics , Physics+ 1 more

Further Mathematics
Physics
Maths

“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

If a car of mass 1000kg travels up a slope inclined at 5 degrees at a speed of 20 meters per second calculate the power output of the car's engine (assuming a resistive force due to friction of 500N)

The curve C has polar equation 'r = 3a(1 + cos(x)). The tangent to C at point A is parallel to the initial line. Find the co-ordinates of A. 0<x<pi

How to use the integrating factor?

What are polar coordinates?

View A Level Further Mathematics tutors

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

mtw:mercury1:status:ok