1. Can you tell me or give me a link where I can learn to take the cube root of a complex number like -9 + 46 i with pencil and paper?
Is it possible to solve it algebraically?

Thanks

2.

3. If you convert the number to polar form, you get a*(e^(i*theta)) with absolute value a and angle i*theta. Then, the cube root is cbrt(a)*e^(i*theta/3). You take the cube root of the radius, and one third of the angle. At least, I think.
OR, you can compute (a + bi)^3 and solve for the real part being -9 and the imaginary part 46. It's a system of equations where you can solve for a and b.
Cube root - Wikipedia, the free encyclopedia gives more detail.

4. For a cube root, there will be 3 solutions. The polar form of will be where is an integer, since is a full rotation and any number of full rotations will give exactly the same polar coordinates.

Then if you do as anticorncob28 suggests, you'll have and then you'll just need to solve for 3 values of to get all your roots.

Keep in mind, needs to be in radians.

5. Thanks for your help, I knew the wiki article, but all you say needs a (good ) calculator.
I think the best way is to proceed like this
x^3-39x = -9:
x=6

180°-((tan^-1 46/-9) = 101°.../3 = 33°.69
cos 33.69 = 0.832

I was wondering how to solve it only with algebra, pencil and paper.
Wolfram gives a very simple factorization in the step-by-step solution, but I ignore the rules.
Thanks again

6. The cube roots of that complex number are going to look pretty hairy. I don't think you'll get exact numbers or even tidy fractions out of that.

There are algorithms for working out square and cube roots on paper (square roots, cube roots), but a calculator would make it easier and you'll probably need a calculator to get . You don't need a fancy calculator. A \$20 scientific calculator will do, or windows calculator has a scientific mode that has all the functions you need.

Do you know how to convert to polar form?

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