Derive the Hicks demand function for $U(x_1,x_2) = x_1^{1/2}x_2^{1/3}$
Let $f(U)=U^{6/5}$. This is a positive monotone transformation of $U$ on $\mathbb{R}_0^+$. So the preferences represented by $U$ are also represented by $V(x_1,x_2):=f(U(x_1,x_2))=x_1^{3/5}x_2^{2/5}$. The utility function $V$ has Cobb-Douglas form and you can use the formula for the Hicksian demand for Cobb-Douglas utilities: $$x_1^*=\left(\frac{3p_2}{2p_1}\right)^{2/5}V,\quad x_2^*=\left(\frac{2p_1}{3p_2}\right)^{3/5}V.$$ Substituting for $V$ you get $$x_1^*=\left(\frac{3p_2}{2p_1}\right)^{2/5}U^{6/5},\quad x_2^*=\left(\frac{2p_1}{3p_2}\right)^{3/5}U^{6/5},$$ or $$x_1^*=\left[\left(\frac{3p_2}{2p_1}\right)U^3\right]^{2/5},\quad x_2^*=\left[\left(\frac{2p_1}{3p_2}\right)U^2\right]^{3/5}.$$