1. Simplify the expression $a^2 (2a^{-1})^2$. Step 1: Write the expression clearly: $$a^2 (2a^{-1})^2$$ Step 2: Apply the power to each factor inside the parentheses: $$(2)^2 (a^{-1})^2 = 4 a^{-2}$$ Step 3: Multiply by $a^2$: $$a^2 \times 4 a^{-2} = 4 a^{2 + (-2)} = 4 a^0$$ Step 4: Recall that $a^0 = 1$ for $a \neq 0$: $$4 \times 1 = 4$$ Final answer: $$4$$ This completes the simplification of the first problem.