?Why does −32=−9$-3^2=-9$ but (−3)2=9$(-3{)}^2=9$

With respect to proof (rather than just a tautology of the mathematical statement like "it's opposite").

Part of it is the notational convention regarding precedence of operations.  When you fully parenthesize  −32$-3^2$  you get  −(32).$-(3^2).$

The other part is the rule of signs.  A negative number times a negative number is a positive number, so  (−3)2$(-3{)}^2$  is a positive number, but of course, −(32)$-(3^2)$  is a negative number.