Which power means are constructible?

For an algebraic number over $\mathbb Q$, in order to be constructable it's minimal polynomial should be of a degree $2^n$ , thus $M_{2^n}$

This is a construction for $M_4$:

$AC=a$ and $AK=b$.

Using a dummy unit length $(AB)$ you construct $GH=a^4$ and $NO=b^4$ (example for $a^2$ construction).

Let $BQ=a^4+b^4$, $R$ is the middle point for $BQ$ so $BR=(a^4+b^4)/2$.

Using the same dummy unit we construct the square root $BT$ of $BR$ and the square root $BW$ of $BU=BT$.

We now have your desired mean. The dummy unit is uninfluent, if you change the lenght of $AB$ the length of $BW$ does not change.