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

Start by noticing that $(2 + \sqrt{5})^{3} = 38 + 17 \sqrt{5}$ and $(2 - \sqrt{5})^{3} = 38 - 17 \sqrt{5}$. Now,
\begin{align}
\sqrt[3]{\ 17\sqrt{5}+38}  - \sqrt[3]{17\sqrt{5}-38} &= \sqrt[3]{(2 + \sqrt{5})^{3}} - \sqrt[3]{(\sqrt{5} - 2)^{3}} \\
&= (2 + \sqrt{5}) - (-2 + \sqrt{5}) \\
&= 4.
\end{align}

What seems to have happened is that most answers presented here follow the pattern:
\begin{align}
\sqrt[3]{\ 17\sqrt{5}+38}  + \sqrt[3]{17\sqrt{5}-38} &= \sqrt[3]{(2 + \sqrt{5})^{3}} + \sqrt[3]{(\sqrt{5} - 2)^{3}} \\
&= (2 + \sqrt{5}) + (-2 + \sqrt{5}) \\
&= 2 \sqrt{5}.
\end{align}
