jmc

Since $a^2$ varies inversely with $b^3$, $(a^2)(b^3)=k$ for some constant $k$. If $a=7$ when $b=3$, then $k=(7^2)(3^3)=(49)(27)=1323$. So if $b=6$, $$\begin{array}{rl}(a^2)(6^3)& =1323\\ 216a^2& =1323\\ \Rightarrow a^2& =\boxed{6.125}\end{array}$$