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On the image ( https://prnt.sc/0jYC5khEFyok), I am stuck. How could I proceed to solve the "E" with blue arrow and derive the consumer expenditure function without solving directly for Hicksian demand ?

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It looks like you basically have it

$\overline{u}=(\frac{1}{4p_x})^{1/4}(\frac{3}{4p_y})^{3/4}*E(p_x,p_y,\overline{u})$

Move everything but $E$ to the left hand side and you have

$$E(p_x,p_y,\overline{u})=\overline{u}\cdot(\frac{1}{4p_x})^{-1/4}\cdot(\frac{3}{4p_y})^{-3/4}$$

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