I made a mistake recently regarding the Gaussian density, by putting the determinant of the variance to the power $\frac{d}{2}$. Would the following argumentation be valid to highlight it should be to the power $\frac{1}{2}$? :
The argument of the exponential must be dimensionless, so whatever unit is $x$ in, the variance matrix entries have dimension $x^2$.
$$ dimension~ ( ~{ x^t \Sigma^{-1} x}) = 0 $$
The density formula is some dimensionless $\alpha$ coming from normalization constant and the exponential, divided by the determinant yields therefore a quantity with dimension $x^{-2*d}$
$$ dimension~ ( \frac{\alpha}{\det \Sigma} ) ~=~ ~x^{-2*d}$$
But we want this to be a density, meaning that multiplied by the volume element $dx$ (of dimension $x^d$) you get a dimensionless number, the "count" of how many elements are in that box
$$ dimension ~( \frac{\alpha}{\det \Sigma}. dx ~) ~=~ 0$$
So one should raise the determinant to the power $\frac{1}{2}$