Consider the utility function $U(x_1,x_2) = x_1^{1/2}x_2^{1/3}$ and the budget line $p_1x_1+p_2x_2 = m$. Then I have to find the Hicks demand function.
I know that to do this I have to solve the following expenditure minimization problem $$ \min_{x_1,x_2} p_1x_1+p_2x_2 $$ subject to $$ x_1^{1/2}x_2^{1/3} = U $$ for some utility level $U$. I have already found the Marshallian demand functions to be $x_1^* = \frac{3m}{5p_1}$ and $x^* = \frac{2m}{5p_2}$.
I thought maybe we could use the well-known Cobb-Douglas demand functions but I don't think we can as $1/2 + 1/3 = 5/6 \neq 1$.
I then tried to use Lagrange and got that
By Lagrange we have that $$ L = p_1x_1+p_2x_2 - \lambda \left( x_1^{1/2}x_2^{1/3}-U \right) $$ Thus the three FOC will be \begin{align} \frac{\partial L}{\partial x_1}: p_1- \lambda \left( \frac{1}{2} x_1^{-1/2}x_2^{1/3} \right) \\ \frac{\partial L}{\partial x_2}: p_2 - \lambda \left( \frac{1}{3} x_1^{1/2}x_2^{-2/3} \right) \\ \frac{\partial L}{\partial \lambda}: x_1^{1/2}x_2^{1/3} = U \end{align} Dividng the first equation with the second equation one gets $$ \frac{3x_2}{2x_1} = \frac{p_1}{p_2} $$ Thus $$ x_1 = \frac{3p_2x_2}{2p_1} $$ Inserting this into $U(x_1,x_2)$ we have that $$ \left(\frac{3p_2x_2}{2p_1} \right)^{1/2} x_2^{1/3} $$ which gives me after some few calculations that $$ x_1^* = \frac{3p_2}{2p_1} \left( \frac{U\sqrt{2}\sqrt{p_1}}{ \sqrt{3} \sqrt{p_2}} \right)^{6/5} $$ $$ x_2^* = \left( \frac{ U \sqrt{2} \sqrt{p_1}}{ \sqrt{3} \sqrt{p_2} } \right)^{1/(5/6)} $$ but this doesn't seem right. Is there an easier solution to this kind of problem?
