Evaluating the real and imaginary parts of a nasty complex number

This seems like an elementary question, but I was unable to find an clear answer to it.

Generally, the real and imaginary parts of a complex number comprised of radicals are not expressible by radicals themselves (see for example Casus irreductibilis, which is a special case where the imaginary part is $0$).

However, a numerical approximation can be done to achieve arbitrary precision for the real and imaginary parts.

What's the general method for evaluating the real and imaginary parts of a complex number?

For an example, see here.

In the above example, an alternative form consisting of trigonometric functions is shown, which suggests that conversion to polar form is a way to go. Is this it?

Is this then also possible on even more complicated, non-algebraic, complex numbers, such as this one?

Ideally what you want to have at the end is a nice complex number of the form : $$z = x+iy$$ Unfortunately as you have shown, complex numbers aren't always written this way... Being able to write them under an exponential form is a great way to evaluate them for a few reasons.

The first reason I see is that it is very easy to take their conjugate form, simply change the sign of the argument.
Another good reason is that $re^{i\theta} = r\cos(\theta) + ir\sin(\theta)$ making it easy to transform into a nice form.
They are very easy to multiply and divide with each another.

One problem though : you can't add them simply.

So when evaluating horrific complex numbers here are my advice:
- If you need to add or subtract complex numbers of the form $x + iy$, then just use the normal formula.
- If you have expression you want to add but aren't of the form $x+iy$, you should normally be able to write them as a polar form and then rewrite them as $x+iy$ using Euler's formula.
- If you want to divide, multiply or take a power of two complex numbers, always use exponential forms, it is the easiest way.

With all of this you should (slowly) be able to evaluate complex numbers written using elementary operations. As you can see, the exponential form is used very often except when adding or subtracting. In the end you should always be able to write it under the form $x+iy$ where the imaginary and real parts are explicit.

Using $x+iy = r e^{i\theta}$ where $r = \sqrt{x^{2}+y^{2}}$ and $\theta = \tan^{-1}(y/x)$ then the expression \begin{align} \phi = \frac{ (11+i\sqrt{3})^{15/71} + (3/7+i\sqrt{41})^{13/21} }{ (13+i\sqrt{47})^{19/11} } \end{align} becomes \begin{align} \phi &= \frac{ r_{1}^{15/71} e^{15\theta_{1}i/71} + r_{2}^{13/21} e^{13\theta_{2}i/21} }{ r_{3}^{19/11} e^{19\theta_{3}i/11} }\\ &= A e^{i\alpha} + B e^{i\beta} \\ &= \left[ A \cos(\alpha) + B \cos(\beta) \right] + i \left[ A \sin(\alpha) + B \sin(\beta) \right]. \end{align} The components are: \begin{align} r_{1} &= 2\sqrt{31} \\ r_{2} &= \frac{1}{7} \sqrt{2018} \\ r_{3} &= 6 \sqrt{6} \\ A &= r_{1}^{15/71} r_{3}^{-19/11} = 2^{-3717/1562} 3^{-57/22} (31)^{15/142} \\ B &= r_{2}^{13/21} r_{3}^{-19/11} = 2^{-1009/462} 3^{-57/22} 7^{-13/21} (1009)^{13/42} \\ \theta_{1} &= \tan^{-1}(\sqrt{3}/11) \\ \theta_{2} &= \tan^{-1}(7\sqrt{41}/3) \\ \theta_{3} &= \tan^{-1}(\sqrt{47}/3) \\ \alpha &= \frac{15}{71} \theta_{1} - \frac{19}{11} \theta_{3} = \frac{15}{71} \tan^{-1}(\sqrt{3}/11) - \frac{19}{11} \tan^{-1}(\sqrt{47}/3) \\ \beta &= \frac{13}{21} \theta_{2} - \frac{19}{11} \theta_{3} = \frac{13}{21} \tan^{-1}(7\sqrt{41}/3) - \frac{19}{11} \tan^{-1}(\sqrt{47}/3). \end{align} The real and imaginary parts of $\phi$ can be calculated by using the appropriate factors listed.