1. Problem: Evaluate $ (2^{-1})(2^{-2}) $. Formula: When multiplying powers with the same base, add the exponents: $ a^m \cdot a^n = a^{m+n} $. Step 1: Apply the rule: $$ (2^{-1})(2^{-2}) = 2^{-1 + (-2)} = 2^{-3} $$ Step 2: Simplify the negative exponent: $$ 2^{-3} = \frac{1}{2^3} = \frac{1}{8} $$ Answer: $ \frac{1}{8} $. 2. Problem: Evaluate $ \left(\frac{4}{3}\right)^{-3} $. Formula: Negative exponent means reciprocal: $$ a^{-n} = \frac{1}{a^n} $$ Step 1: Apply the negative exponent: $$ \left(\frac{4}{3}\right)^{-3} = \left(\frac{3}{4}\right)^3 $$ Step 2: Cube numerator and denominator: $$ \left(\frac{3}{4}\right)^3 = \frac{3^3}{4^3} = \frac{27}{64} $$ Answer: $ \frac{27}{64} $. 3. Problem: Evaluate $ \left(3^{-2} \cdot 5^2\right)^{-2} $. Formula: Power of a product: $$ (ab)^n = a^n b^n $$ Step 1: Simplify inside the parentheses: $$ 3^{-2} \cdot 5^2 = \frac{1}{3^2} \cdot 5^2 = \frac{5^2}{3^2} = \frac{25}{9} $$ Step 2: Apply the outer exponent: $$ \left(\frac{25}{9}\right)^{-2} = \left(\frac{9}{25}\right)^2 = \frac{9^2}{25^2} = \frac{81}{625} $$ Answer: $ \frac{81}{625} $. 4. Problem: Evaluate $ \left(\frac{5}{3^{-2}}\right)^{-3} $. Step 1: Simplify the denominator: $$ 3^{-2} = \frac{1}{3^2} = \frac{1}{9} $$ Step 2: Rewrite the fraction: $$ \frac{5}{3^{-2}} = 5 \cdot 3^2 = 5 \cdot 9 = 45 $$ Step 3: Apply the negative exponent: $$ 45^{-3} = \frac{1}{45^3} $$ Answer: $ \frac{1}{45^3} $. 5. Problem: Evaluate $ \left(\frac{3^{-2}}{2^{-3}}\right)^{-3} $. Step 1: Simplify numerator and denominator: $$ 3^{-2} = \frac{1}{3^2} = \frac{1}{9}, \quad 2^{-3} = \frac{1}{2^3} = \frac{1}{8} $$ Step 2: Form the fraction: $$ \frac{3^{-2}}{2^{-3}} = \frac{\frac{1}{9}}{\frac{1}{8}} = \frac{1}{9} \cdot \frac{8}{1} = \frac{8}{9} $$ Step 3: Apply the outer exponent: $$ \left(\frac{8}{9}\right)^{-3} = \left(\frac{9}{8}\right)^3 = \frac{9^3}{8^3} = \frac{729}{512} $$ Answer: $ \frac{729}{512} $.