Calculate $x^3 + \frac{1}{x^3}$ Question
$x^2 + \frac{1}{x^2}=34$ and $x$ is a natural number. Find the value of $x^3 +  \frac{1}{x^3}$ and choose the correct answer from the following options:

*

*198

*216

*200

*186


What I have did yet
I tried to find the value of $x + \frac{1}{x}$. Here are my steps to do so:
$$x^2 + \frac{1}{x^2} = 34$$
$$\text{Since}, (x+\frac{1}{x})^2 = x^2 + 2 + \frac{1}{x^2}$$
$$\Rightarrow (x+\frac{1}{x})^2-2=34$$
$$\Rightarrow (x+\frac{1}{x})^2=34+2 = 36$$
$$\Rightarrow x+\frac{1}{x}=\sqrt{36}=6$$
I have calculated the value of $x+\frac{1}{x}$ is $6$. I do not know what to do next. Any help will be appreciated. Thank you in advanced!

Let have a look at this: finding $x^6+y^6$ given $x+y$ and $xy$
$$U_n=x^n+\frac 1{x^n}$$
$s=x+\frac 1x=U_1$ and $p=x\times\frac 1x=1$
also we have $\begin{cases}U_2=34\\U_0=x^0+\frac 1{x^0}=1+1=2\end{cases}$
We have the relation $$U_{n+1}=U_1U_n-U_{n-1}$$
$U_2=U_1^2-U_0\implies U_1^2=34+2=36\implies U_1=\pm 6$
And $U_3=U_1U_2-U_1=\pm 6(34-1)=\pm 198\quad$ (among proposed answers the positive one fits).
Note: $x$ cannot be integer, since $x+\frac 1x=\pm 6\iff x=\pm(3\pm 2\sqrt{2})$
This method can be generalized to most $x^n+y^n$ types of questions.
