Symmetry group is cocentral in similitude group for nondegenerate bilinear form if nth power map on squares is surjective

Statement
Suppose $$k$$ is a field, $$V$$ a finite-dimensional vector space over $$k$$, and $$b$$ a nondegenerate bilinear form on $$V$$. Suppose that the map $$x \mapsto x^n$$, when restricted to the squares in the multiplicative group of $$k$$, is surjective (i.e., every square is the $$n^{th}$$ power of a square). Then, the symmetry group for $$b$$ is a fact about::cocentral subgroup of the similitude group for $$b$$.