27. \( \frac{15^{7} \times 8^{6}}{5^{6} \times 24} \) 28. \( \frac{2^{2005} \times 5^{2000}}{100^{1000}} \) 29. \( \frac{7^{2 x}-1}{3 \times 7^{x}+3} \) 30. \( \frac{16^{x}-1}{5 \times 4^{x}-5} \) 31. \( \frac{25^{x}-4}{2 \times 5^{x}-4} \) 32. \( \frac{9^{x}-3^{x}-2}{6^{x}-2^{x+1}} \)

Below are the step‐by‐step simplifications: 27. Simplify (15⁷ × 8⁶)/(5⁶ × 24).  • Write 15 and 8 as products of primes:   – 15 = 3·5, so 15⁷ = 3⁷ × 5⁷.   – 8 = 2³, so 8⁶ = (2³)⁶ = 2¹⁸.  • Factor the denominator:   – 24 = 3 × 8 = 3 × 2³, so 5⁶ × 24 = 5⁶ × 3 × 2³.  • Thus,    (15⁷ × 8⁶)/(5⁶ × 24) = (3⁷ × 5⁷ × 2¹⁸)/(5⁶ × 3 × 2³)  • Cancel common factors:   – For 3: 3⁷/3 = 3⁶.   – For 5: 5⁷/5⁶ = 5.   – For 2: 2¹⁸/2³ = 2¹⁵.  • The simplified expression is 3⁶ × 5 × 2¹⁵. 28. Simplify (2^2005 × 5^2000)/(100^1000).  • Rewrite 100 as 10². Then 100^1000 = (10²)^1000 = 10²⁰⁰0.  • Since 10 = 2×5, we have 10²⁰⁰0 = 2²⁰⁰0 × 5²⁰⁰0.  • Now,   (2^2005 × 5^2000)/(2^2000 × 5^2000) = 2^(2005–2000)= 2⁵ = 32. 29. Simplify (7^(2x) – 1)/(3×7^x + 3).  • Notice the numerator is a difference of squares: 7^(2x) – 1 = (7^x – 1)(7^x + 1).  • Factor the denominator: 3×7^x + 3 = 3(7^x + 1).  • Cancel the common factor (7^x + 1) (assuming 7^x ≠ –1):   Result = (7^x – 1)/3. 30. Simplify (16^x – 1)/(5×4^x – 5).  • Write 16^x in terms of 4^x: 16^x = (4²)^x = 4^(2x). Then numerator = 4^(2x) – 1.  • Recognize the difference of squares: 4^(2x) – 1 = (4^x – 1)(4^x + 1).  • Factor the denominator: 5×4^x – 5 = 5(4^x – 1).  • Cancel (4^x – 1) (provided 4^x ≠ 1):   Result = (4^x + 1)/5. 31. Simplify (25^x – 4)/(2×5^x – 4).  • Write 25^x as (5²)^x = 5^(2x). So numerator = 5^(2x) – 4.  • Notice that 5^(2x) – 4 is a difference of squares: 5^(2x) – 4 = (5^x)² – 2² = (5^x – 2)(5^x + 2).  • Factor the denominator: 2×5^x – 4 = 2(5^x – 2).  • Cancel the common factor (5^x – 2) (assuming 5^x ≠ 2):   Result = (5^x + 2)/2. 32. Simplify (9^x – 3^x – 2)/(6^x – 2^(x+1)).  • Express all terms in prime bases:   – 9^x = (3²)^x = 3^(2x).   – 6^x = (2×3)^x = 2^x×3^x.   – 2^(x+1) = 2×2^x.  • Rewrite the numerator with t = 3^x:   – Numerator becomes t² – t – 2, which factors as (t – 2)(t + 1); that is, (3^x – 2)(3^x + 1).  • Rewrite the denominator:   – Denom = 2^x×3^x – 2×2^x = 2^x(3^x – 2).  • Cancel (3^x – 2) (assuming 3^x ≠ 2):   Result = (3^x + 1)/(2^x). These are the simplified forms for problems 27–32.