Minimal polynomial of $\sqrt{2}+\sqrt[3]{3}$.

There is a standard procedure for computing the minimal polynomial for the sum of two numbers from their minimal polynomials.

Let $f(X)$ and $g(Y)$ be the minimal polynomials of the algebraic numbers $\alpha$ and $\beta$. Set $Z=X+Y$. Now consider the polynomial $f(Z-Y)$ and eliminate from this the variable $Y$ using the relation $g(Y)=0$. Then thhe resulting polynomial involving just $Z$ is the minimal polynomial for $\alpha+\beta$.

Here it boils down to eliminating $Y$ from $ (Z-Y)^2-2$ using $Y^3-3=0$.