Can you give an example for $f,g\in\mathbb K[x,y]$ and $\sqrt{(f^2,g^3)}\neq(f,g)$ where $(.)$ means the ideal
In general how do you perform computations with ideals, if $f$ and $g$ were monomials then the equality should hold, $\sqrt{(x^2,y^3)}=\sqrt{(x^2)(y^3)}=\sqrt{(x^2)}\sqrt{(y^3)}=(x)(y)=(x,y)$ am I correct ? just usual multiplication ?
Take $f=x^6, g=y^6$. Then the radical ideal will have $x$ and $y$ in it. However the ideal generated by $x^3$ and $y^2$ will not have them.