# Transforming an indirect utility function into an expenditure function

in preparation for our upcoming exam, our tutor has uploaded sample questions. One of them is the following:

This is my solution:

Which differs from the solution, which gives the ordinary square root of the same parameters, not the 4th-one. I can't find my mistake, does anyone have any ideas?

An indirect utility function should satisfy $$v(\alpha p_1,\alpha p_2,\alpha y)=v(p_1,p_2,y)$$. In other words, if income goes up and prices go up by the same factor then everyone is as well off as they were before.
Unfortunately, in this case $$v(\alpha p_1, \alpha p_2, \alpha y)= \frac{1}2\alpha^2\frac{y^4}{p_1p_2}$$