Exercise 5.7 \( \begin{array}{ll}\text { Calculate the following: } & \text { 2. } 5^{3} \times 7^{-2} \times 5^{-3} \times 7^{4} \\ \text { 1. } 2^{2} \times 3 \times 2^{-2} \times 3^{-1} \\ \text { (3) } 2^{3} \times 11^{-4} \times 2^{-2} \times 11^{3} & \text { (4. } 2^{-1} \times 6^{2} \times 3^{-2} \\ \text { (5) } 3^{-3} \times 9 \times 3^{-2} \perp & \text { 8. } 9^{-2} \times 2^{2} \times 3^{-1} \\ \text { 7. } 9^{-1} \times 6^{3} \times 4^{-2} & \text { 10. } 2^{-1} \times 6^{3} \div 3^{2} \\ \text { 9. } 5^{-1} \div 5 & \end{array} \)

Below are step‐by‐step solutions for each item. ────────────────────────────── 1) Calculate 2² × 3 × 2⁻² × 3⁻¹ • Group the factors by base:   – For base 2: 2² × 2⁻² = 2^(2 – 2) = 2⁰ = 1.   – For base 3: 3¹ × 3⁻¹ = 3^(1 – 1) = 3⁰ = 1. • Multiply the results: 1 × 1 = 1. Answer: 1 ────────────────────────────── 2) Calculate 5³ × 7⁻² × 5⁻³ × 7⁴ • Group by base:   – For base 5: 5³ × 5⁻³ = 5^(3 – 3) = 5⁰ = 1.   – For base 7: 7⁻² × 7⁴ = 7^(–2 + 4) = 7² = 49. • Multiply: 1 × 49 = 49. Answer: 49 ────────────────────────────── 3) Calculate 2³ × 11⁻⁴ × 2⁻² × 11³ • Group by base:   – For base 2: 2³ × 2⁻² = 2^(3 – 2) = 2¹ = 2.   – For base 11: 11⁻⁴ × 11³ = 11^(–4 + 3) = 11^(–1) = 1/11. • Multiply: 2 × (1/11) = 2/11. Answer: 2/11 ────────────────────────────── 4) Calculate 2⁻¹ × 6² × 3⁻² Method 1: Compute directly • Evaluate 6² = 36. • Then: 2⁻¹ × 36 = 36/2 = 18. • Finally, 18 × 3⁻² = 18 × (1/9) = 2. Method 2: Express 6 in terms of its prime factors • Note 6 = 2 × 3 so that 6² = 2² × 3². • Now: 2⁻¹ × (2² × 3²) × 3⁻² = 2^(–1+2) × 3^(2–2) = 2¹ × 3⁰ = 2 × 1 = 2. Answer: 2 ────────────────────────────── 5) Calculate 3⁻³ × 9 × 3⁻² • Write 9 as 3². • Then: 3⁻³ × 3² × 3⁻² = 3^(–3 + 2 – 2) = 3^(–3) = 1/27. Answer: 1/27 ────────────────────────────── 7) Calculate 9⁻¹ × 6³ × 4⁻² • Evaluate each term:   – 9⁻¹ = 1/9.   – 6³ = 216.   – 4⁻² = 1/(4²) = 1/16. • Multiply: (1/9) × 216 = 216/9 = 24. • Then: 24 × (1/16) = 24/16 = 3/2. Answer: 3/2 ────────────────────────────── 8) Calculate 9⁻² × 2² × 3⁻¹ • Express 9 as 3² so that 9⁻² = (3²)⁻² = 3^(–4). • Now combine the 3’s: 3^(–4) × 3⁻¹ = 3^(–5). • The 2’s remain: 2² = 4. • Thus, the product = 4 × 3^(–5) = 4/(3⁵). • Since 3⁵ = 243, the answer is 4/243. Answer: 4/243 ────────────────────────────── 9) Calculate 5⁻¹ ÷ 5 • Recall that division by 5 is the same as multiplying by 1/5, and note that 5 = 5¹. • Thus, 5⁻¹ ÷ 5¹ = 5^(–1 – 1) = 5⁻² = 1/(5²) = 1/25. Answer: 1/25 ────────────────────────────── 10) Calculate 2⁻¹ × 6³ ÷ 3² Method 1: Compute directly • Evaluate 6³ = 216 and 2⁻¹ = 1/2. • Then: (1/2 × 216) = 108. • Divide by 3², which is 9: 108 ÷ 9 = 12. Method 2: Express factors in prime factorization • Write 6 as 2 × 3, so 6³ = 2³ × 3³. • Then: 2⁻¹ × (2³ × 3³) = 2^(–1+3) × 3³ = 2² × 3³ = 4 × 27 = 108. • Now, dividing by 3²: 108 ÷ 9 = 12. Answer: 12 ────────────────────────────── Summary of Answers: 1) 1 2) 49 3) 2/11 4) 2 5) 1/27 7) 3/2 8) 4/243 9) 1/25 10) 12