1. Problem: Simplify each expression with positive exponents only and evaluate when possible. 2. a) Simplify $ (5^{-3} \times 5^{9})(5^{3}) $. - Use the product of powers rule: $ a^{m} \times a^{n} = a^{m+n} $. - Calculate inside the parentheses: $ 5^{-3+9} = 5^{6} $. - Multiply by $ 5^{3} $: $ 5^{6} \times 5^{3} = 5^{6+3} = 5^{9} $. - Evaluate $ 5^{9} = 1953125 $. 3. b) Simplify $ (x^{5} y^{6})(x^{3} y^{7}) $. - Apply product of powers for each variable: $ x^{5} \times x^{3} = x^{8} $ and $ y^{6} \times y^{7} = y^{13} $. - Result: $ x^{8} y^{13} $. 4. c) Simplify $ \left(\frac{2}{3}\right)^{-2} \times 2^{-1} $. - Use negative exponent rule: $ a^{-n} = \frac{1}{a^{n}} $. - $ \left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^{2} = \frac{9}{4} $. - $ 2^{-1} = \frac{1}{2} $. - Multiply: $ \frac{9}{4} \times \frac{1}{2} = \frac{9}{8} $. 5. d) Simplify $ \frac{(x^{5} y)^{2}}{x^{7} y^{2}} $. - Apply power to product: $ (x^{5})^{2} y^{2} = x^{10} y^{2} $. - Divide: $ \frac{x^{10} y^{2}}{x^{7} y^{2}} = x^{10-7} y^{2-2} = x^{3} y^{0} $. - Since $ y^{0} = 1 $, result is $ x^{3} $. 6. e) Simplify $ \frac{(6 \times 5)^{3}}{6^{3}} $. - Calculate numerator: $ (30)^{3} = 27000 $. - Calculate denominator: $ 6^{3} = 216 $. - Divide: $ \frac{27000}{216} = 125 $. 7. f) Simplify $ (6 \times 3)^{3} $. - Calculate inside parentheses: $ 18^{3} = 5832 $. 8. g) Simplify $ (n)(2)(3) $. - Multiply constants: $ 2 \times 3 = 6 $. - Result: $ 6n $. 9. h) Simplify $ (n^{2})(w)(p) $. - Result: $ n^{2} w p $. 10. i) Simplify $ 9 + 9U = (3 \times 3)(U + 9) $. - Factor 9 as $ 3 \times 3 $. - Distribute: $ (3 \times 3)(U + 9) = 9U + 81 $. - Note: $ 9 + 9U \neq 9U + 81 $, so the equality is incorrect as written. 11. j) Expression $ 24 = $ is incomplete; no simplification possible. Final answers: a) $1953125$ b) $x^{8} y^{13}$ c) $\frac{9}{8}$ d) $x^{3}$ e) $125$ f) $5832$ g) $6n$ h) $n^{2} w p$ i) Expression incorrect as stated j) Incomplete expression