(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I need to find the solution to [tex](2-11i)^{\frac{1}{3}}[/tex]

2. Relevant equations

If [tex](2-11i)^{\frac{1}{3}}[/tex] were to equal (a + bi) for some real numbers a and b then [tex]2 - 11i = a^3 +3a^2bi-3ab^2-b^3i[/tex]

3. The attempt at a solution

From above [tex]a^3-3ab^2 = 2[/tex] and [tex]3a^2b - b^3 = -11[/tex]

I can factorise (but only slightly) as follows:

[tex]a(a^2-b^2) = 2 + 2ab^2[/tex]

[tex]b(a^2-b^2) = -11 - 2a^2b[/tex]

after losing the a^2-b^2 I'm left with [tex]2b + 2ab^3 = -11a - 2a^3b[/tex] and I can see no other useful factorisations or substitutions :(

The actual values I need to find here are simple and with not so much guess work found that a = 2 and b = -1. My problem is that I'm not so sure that guess-work is the correct method to be using. I could probably plot both functions and find where there is a point of intersection but is there an algebraic method I can employ?...If so can anyone throw me any pointers?

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# 3rd root of a complex number

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