<s>[INST] Find the value of $\sqrt[4]{\alpha}-\sqrt[4]{\beta}$,where $\sqrt[4]{.}$ denotes the principal value. If $\alpha$ and $\beta$ are the roots of the equation $x^2-34x+1=0$,find the value of $\sqrt[4]{\alpha}-\sqrt[4]{\beta}$,where $\sqrt[4]{.}$ denotes the principal value.

I found out the $\alpha$ and $\beta$.
$\alpha,\beta=\frac{34\pm\sqrt{32\times 36}}{2}=17\pm12\sqrt2$ but i do not know how to find $\sqrt[4]{\alpha}-\sqrt[4]{\beta}$.
 [/INST] One may observe that
$$
(\sqrt{2}-1)^2=3-2\sqrt{2}
$$ and that
$$
(3-2\sqrt{2})^2=17-12\sqrt{2}
$$ thus
$$
(\sqrt{2}-1)^4=17-12\sqrt{2}
$$ giving

$$
\sqrt[4]{\alpha}=\sqrt[4]{17-12\sqrt{2}}=\sqrt{2}-1,
$$ 

similarly

$$
\sqrt[4]{\beta}=\sqrt[4]{17+12\sqrt{2}}=\sqrt{2}+1.
$$

</s>