Here's how to simplify the expressions: a) Simplify $4^2 \times 4^{-3}$ Step 1: Use the exponent rule $a^m \times a^n = a^{m+n}$. $$4^2 \times 4^{-3} = 4^{2 + (-3)}$$ $$4^{2 + (-3)} = 4^{2-3}$$ $$4^{2-3} = 4^{-1}$$ Step 2: Convert the negative exponent to a positive exponent using the rule $a^{-n} = \frac{1}{a^n}$. $$4^{-1} = \frac{1}{4^1}$$ $$4^{-1} = \frac{1}{4}$$ The simplified expression is $\boxed{\frac{1}{4}}$. b) Simplify $\frac{3^2 (2^2)^{-2}}{2^3}$ Step 1: Simplify the term $(2^2)^{-2}$ in the numerator using the exponent rule $(a^m)^n = a^{mn}$. $$(2^2)^{-2} = 2^{2 \times (-2)}$$ $$(2^2)^{-2} = 2^{-4}$$ Step 2: Substitute this back into the expression. $$\frac{3^2 \times 2^{-4}}{2^3}$$ Step 3: Simplify the terms with base 2 using the exponent rule $\frac{a^m}{a^n} = a^{m-n}$. $$\frac{2^{-4}}{2^3} = 2^{-4-3}$$ $$2^{-4-3} = 2^{-7}$$ Step 4: Combine the simplified terms. $$3^2 \times 2^{-7}$$ Step 5: Convert the negative exponent to a positive exponent using the rule $a^{-n} = \frac{1}{a^n}$. $$2^{-7} = \frac{1}{2^7}$$ Step 6: Write the expression with positive exponents. $$3^2 \times \frac{1}{2^7} = \frac{3^2}{2^7}$$ Step 7: Calculate the values of the powers. $$3^2 = 9$$ $$2^7 = 128$$ The simplified expression is $\boxed{\frac{9}{128}}$.