Solve cos((1/3)*arccos(x))

[Zetison]

Yes, and this is what I get. I guess that's the far as I am going to get:
$$cos(\frac{1}{3}arccos(x)) = \frac{(x + \sqrt{x^2-1})^{1/3}}{2} + \frac 1 {2(x+\sqrt{x^2-1})^{1/3}}$$

But again, my x is defined as

$$x = \frac{473419}{6121\sqrt{6121}}$$

approximate $$x = 0.9885806704$$.

So my final goal here is to get exact values for

$$z = \frac{79}{60} + \frac{1}{30} \sqrt{6121} cos(\frac{1}{3}arccos(x)) = \frac{79}{60} + \frac{1}{30} \sqrt{6121} \frac{(x + \sqrt{x^2-1})^{1/3}}{2} + \frac 1 {2(x+\sqrt{x^2-1})^{1/3}}$$

But that is very difficult to express when
$$x = \frac{473419}{6121\sqrt{6121}}$$.

Is it possible?

 

[Zetison]

The problem of this thread is actually solved. Maybe I shall make a new thread about my main goal...

 

[Zetison]

See you can do it! Take the solution with the positive root and plug it into the exponential form of your original equation.

Why are we not interested in the negative root?