Question: 8. Given that $2^{2x - 1} = \frac{\sqrt{8}}{4}$, find the value of $x$. 9. Given that $2(16^{2m + 1}) = 8$, find the value of $m$. 10. (a) Simplify $(p^3)^5 \div p^7 \times \frac{1}{p^2}$. (b) Given that $\frac{3^n \times 27}{(3^n)^2} = 81$, find the value of $n$. (c) Given that $\left(\frac{x}{y}\right)^{n - 1} = \left(\frac{y}{x}\right)^{n - 5}$, find the value of $n$. 11. (a) Simplify $\frac{\sqrt{m^4 n} \times (m n^3)^2}{m n^{5/2}}$. (b) Simplify $\frac{18^{n + 1} \times 3^{1 - n}}{2^{n - 1}}$. (c) Given that $16(25^x) = 125(32^y)$, find the values of $x$ and $y$. Teach me this??

1. **Problem 8:** Solve $2^{2x - 1} = \frac{\sqrt{8}}{4}$. 2. Express $\sqrt{8}$ as $8^{1/2} = (2^3)^{1/2} = 2^{3/2}$. 3. Rewrite right side: $\frac{2^{3/2}}{4} = \frac{2^{3/2}}{2^2} = 2^{3/2 - 2} = 2^{-1/2}$. 4. So equation is $2^{2x - 1} = 2^{-1/2}$. 5. Since bases equal, exponents equal: $2x - 1 = -\frac{1}{2}$. 6. Solve for $x$: $2x = 1 - \frac{1}{2} = \frac{1}{2}$, so $x = \frac{1}{4}$. --- 7. **Problem 9:** Solve $2(16^{2m + 1}) = 8$. 8. Express $16$ as $2^4$, so $16^{2m + 1} = (2^4)^{2m + 1} = 2^{4(2m + 1)} = 2^{8m + 4}$. 9. Equation becomes $2 \times 2^{8m + 4} = 8$ or $2^{1 + 8m + 4} = 2^3$. 10. Simplify exponent: $1 + 8m + 4 = 8m + 5$. 11. So $2^{8m + 5} = 2^3$ implies $8m + 5 = 3$. 12. Solve for $m$: $8m = 3 - 5 = -2$, so $m = -\frac{1}{4}$. --- 13. **Problem 10(a):** Simplify $(p^3)^5 \div p^7 \times \frac{1}{p^2}$. 14. Use power of a power: $(p^3)^5 = p^{15}$. 15. Expression is $p^{15} \div p^7 \times p^{-2} = p^{15} \times p^{-7} \times p^{-2}$. 16. Combine exponents: $15 - 7 - 2 = 6$. 17. Final answer: $p^6$. --- 18. **Problem 10(b):** Solve $\frac{3^n \times 27}{(3^n)^2} = 81$. 19. Express $27 = 3^3$ and $81 = 3^4$. 20. Denominator: $(3^n)^2 = 3^{2n}$. 21. Expression: $\frac{3^n \times 3^3}{3^{2n}} = \frac{3^{n+3}}{3^{2n}} = 3^{n+3 - 2n} = 3^{3 - n}$. 22. Set equal: $3^{3 - n} = 3^4$. 23. Equate exponents: $3 - n = 4$. 24. Solve for $n$: $-n = 1$, so $n = -1$. --- 25. **Problem 10(c):** Solve $\left(\frac{x}{y}\right)^{n - 1} = \left(\frac{y}{x}\right)^{n - 5}$. 26. Rewrite right side: $\left(\frac{y}{x}\right)^{n - 5} = \left(\frac{x}{y}\right)^{-(n - 5)} = \left(\frac{x}{y}\right)^{5 - n}$. 27. So equation is $\left(\frac{x}{y}\right)^{n - 1} = \left(\frac{x}{y}\right)^{5 - n}$. 28. Since bases equal and nonzero, exponents equal: $n - 1 = 5 - n$. 29. Solve: $2n = 6$, so $n = 3$. --- 30. **Problem 11(a):** Simplify $\frac{\sqrt{m^4 n} \times (m n^3)^2}{m n^{5/2}}$. 31. Simplify $\sqrt{m^4 n} = (m^4 n)^{1/2} = m^{4 \times 1/2} n^{1/2} = m^2 n^{1/2}$. 32. Simplify $(m n^3)^2 = m^2 n^{6}$. 33. Numerator: $m^2 n^{1/2} \times m^2 n^6 = m^{2+2} n^{1/2 + 6} = m^4 n^{13/2}$. 34. Denominator: $m n^{5/2}$. 35. Expression: $\frac{m^4 n^{13/2}}{m n^{5/2}} = m^{4-1} n^{13/2 - 5/2} = m^3 n^{8/2} = m^3 n^4$. --- 36. **Problem 11(b):** Simplify $\frac{18^{n + 1} \times 3^{1 - n}}{2^{n - 1}}$. 37. Express $18 = 2 \times 3^2$. 38. So $18^{n+1} = (2 \times 3^2)^{n+1} = 2^{n+1} 3^{2(n+1)} = 2^{n+1} 3^{2n + 2}$. 39. Expression becomes $\frac{2^{n+1} 3^{2n + 2} \times 3^{1 - n}}{2^{n - 1}} = \frac{2^{n+1}}{2^{n-1}} \times 3^{2n + 2 + 1 - n}$. 40. Simplify powers of 2: $2^{(n+1) - (n-1)} = 2^{2}$. 41. Simplify powers of 3: $3^{(2n + 2 + 1 - n)} = 3^{n + 3}$. 42. Final answer: $2^2 3^{n+3} = 4 \times 3^{n+3}$. --- 43. **Problem 11(c):** Solve $16 (25^x) = 125 (32^y)$. 44. Express bases as powers of primes: $16 = 2^4$, $25 = 5^2$, $125 = 5^3$, $32 = 2^5$. 45. Rewrite: $2^4 (5^{2x}) = 5^3 (2^{5y})$. 46. Rearrange: $2^4 5^{2x} = 5^3 2^{5y}$. 47. Group by base: $2^4 / 2^{5y} = 5^3 / 5^{2x}$. 48. Simplify exponents: $2^{4 - 5y} = 5^{3 - 2x}$. 49. Since bases 2 and 5 are different primes, both sides equal only if both equal 1. 50. So $4 - 5y = 0$ and $3 - 2x = 0$. 51. Solve: $4 = 5y \Rightarrow y = \frac{4}{5}$, $3 = 2x \Rightarrow x = \frac{3}{2}$. --- **Final answers:** - $x = \frac{1}{4}$ - $m = -\frac{1}{4}$ - (10a) $p^6$ - (10b) $n = -1$ - (10c) $n = 3$ - (11a) $m^3 n^4$ - (11b) $4 \times 3^{n+3}$ - (11c) $x = \frac{3}{2}$, $y = \frac{4}{5}$