<s>[INST] Using Cardano's method to find an algebraic equation whose root is $\sqrt{2} +\sqrt[3]{3}$ $\sqrt{2} +\sqrt[3]{3}$ solution of an algebraic equation $ a_{0}x^{n}+a_{1}x^{n-1}+a_{2}x^{n-2}+\cdots+a_{0}=0 $
How to find this equation?
I tried Cardano's method, noting that 
$$\sqrt{2} +\sqrt[3]{3} = \sqrt[3]{\sqrt{8}} +\sqrt[3]{3}$$ 
It means
$$\begin{align}
-\frac{q}{2}+\sqrt{ \frac{q^{2}}{4}+\frac{p^{3}}{27}}&=3 \\[4pt]
-\frac{q}{2}-\sqrt{ \frac{q^{2}}{4}+\frac{p^{3}}{27}}&=\sqrt{8} 
\end{align}$$
but this system doesn't have natural solutions.
Maybe it has rational $p$ and $q$?  
 [/INST] In the same spirit as Awe Kumar Jha, let $x=\sqrt 2 + \sqrt[3]{3}$, that is to say
$$x-\sqrt 2 =\sqrt[3]{3}\implies (x-\sqrt 2) ^3=3$$ Expand and group terms to get
$$x^3-3 \sqrt{2} x^2+6 x-(2 \sqrt{2}+3)=0$$ Use Cardano method; it will show that this is the only real root of the equation.
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