Value of an expression with cube root radical What is the value of the following expression?
$$\sqrt[3]{\ 17\sqrt{5}+38}  - \sqrt[3]{17\sqrt{5}-38}$$

\begin{align} 
x&=\sqrt[3]{17\sqrt{5}+38}  - \sqrt[3]{17\sqrt{5}-38}
\end{align}
Note that 
\begin{align}
17\sqrt{5}-38&=\frac{1}{17\sqrt{5}+38}.
\end{align}
Let $a=\sqrt[3]{17\sqrt{5}+38}$.
Then we have 
\begin{align}
x^3&=\left(a-\frac1a\right)^3
\\
x^3&= a^3-3a+\frac3a-\frac{1}{a^3}
\\
x^3+3\left(a-\frac1a\right)&= a^3-\frac{1}{a^3}
\\
x^3+3x-76&=0
\\
(x-4)(x^2+4x+19)=0,
\end{align}
The only real solution $x=4$ is the answer.
