1. **Problem 15:** Solve the equation $((0.1(6))^{3x} - 5 = 1296)$. 2. **Problem 16:** Find how much the root of the equation $3^{x+1} \cdot 27^{x-1} = 9^7$ is less than 10. 3. **Problem 17:** Find how much the root of the equation $(\frac{2}{3})^x = \sqrt[4]{1.5}$ is less than 1. 4. **Problem 18:** Solve the equation $\frac{2^{2x - 1} \cdot 4^{x+1}}{8^{x-1}} = 64$. 5. **Problem 19:** Solve the equation $0.125 \cdot 4^{2x-3} = \left(\frac{\sqrt{2}}{8}\right)^{-x}$. 6. **Problem 20:** Solve the equation $2^{x-5} + 2^x = 64$. 7. **Problem 21:** Find the product of roots of the equation $2^x \cdot x^2 - 2x^2 + 2 - 2^x = 0$. 8. **Problem 22:** Solve the equation $3^2 \cdot 3^4 \cdot 3^6 \cdots 3^{2n} = (\frac{1}{81})^{-5}$. 9. **Problem 23:** Solve the equation $5^2 \cdot 5^4 \cdot 5^6 \cdots 5^{2n} = 0.04^{-28}$. 10. **Problem 24:** Find the product of roots of the equation $2x^2 - 6x - 5 = 16\sqrt{2}$. --- ### Step-by-step solutions: **15.** Given $((0.1(6))^{3x} - 5 = 1296)$, assuming $0.1(6)$ means $1.6$: 1. Write equation: $1.6^{3x} - 5 = 1296$ 2. Add 5 to both sides: $1.6^{3x} = 1301$ 3. Take logarithm: $3x \log(1.6) = \log(1301)$ 4. Solve for $x$: $x = \frac{\log(1301)}{3 \log(1.6)}$ 5. Calculate approximate value: $x \approx 3$ Answer: C) 3 **16.** Given $3^{x+1} \cdot 27^{x-1} = 9^7$: 1. Express all bases as powers of 3: $27 = 3^3$, $9 = 3^2$ 2. Rewrite: $3^{x+1} \cdot (3^3)^{x-1} = (3^2)^7$ 3. Simplify exponents: $3^{x+1} \cdot 3^{3x-3} = 3^{14}$ 4. Combine: $3^{4x - 2} = 3^{14}$ 5. Equate exponents: $4x - 2 = 14 \Rightarrow 4x = 16 \Rightarrow x = 4$ 6. Difference from 10: $10 - 4 = 6$ Answer: D) 6 **17.** Given $(\frac{2}{3})^x = \sqrt[4]{1.5}$: 1. Rewrite right side: $1.5^{1/4}$ 2. Take logarithm: $x \log(\frac{2}{3}) = \frac{1}{4} \log(1.5)$ 3. Solve for $x$: $x = \frac{\frac{1}{4} \log(1.5)}{\log(\frac{2}{3})}$ 4. Calculate approximate value: $x \approx -0.75$ 5. Difference from 1: $1 - (-0.75) = 1.75$ Answer: A) 1.75 **18.** Given $\frac{2^{2x - 1} \cdot 4^{x+1}}{8^{x-1}} = 64$: 1. Express all bases as powers of 2: $4 = 2^2$, $8 = 2^3$, $64 = 2^6$ 2. Rewrite: $\frac{2^{2x - 1} \cdot (2^2)^{x+1}}{(2^3)^{x-1}} = 2^6$ 3. Simplify exponents: $\frac{2^{2x - 1} \cdot 2^{2x + 2}}{2^{3x - 3}} = 2^6$ 4. Combine numerator: $2^{4x + 1} / 2^{3x - 3} = 2^6$ 5. Simplify: $2^{4x + 1 - 3x + 3} = 2^6 \Rightarrow 2^{x + 4} = 2^6$ 6. Equate exponents: $x + 4 = 6 \Rightarrow x = 2$ Answer: B) 2 **19.** Given $0.125 \cdot 4^{2x-3} = \left(\frac{\sqrt{2}}{8}\right)^{-x}$: 1. Express constants as powers of 2: $0.125 = 2^{-3}$, $4 = 2^2$, $8 = 2^3$ 2. Rewrite: $2^{-3} \cdot (2^2)^{2x-3} = \left(\frac{2^{1/2}}{2^3}\right)^{-x}$ 3. Simplify: $2^{-3} \cdot 2^{4x - 6} = (2^{1/2 - 3})^{-x} = 2^{(5/2)x}$ 4. Combine left side: $2^{4x - 9} = 2^{(5/2)x}$ 5. Equate exponents: $4x - 9 = \frac{5}{2}x$ 6. Solve: $4x - \frac{5}{2}x = 9 \Rightarrow \frac{3}{2}x = 9 \Rightarrow x = 6$ Answer: E) 6 **20.** Given $2^{x-5} + 2^x = 64$: 1. Express 64 as power of 2: $64 = 2^6$ 2. Factor $2^{x-5}$: $2^{x-5} + 2^x = 2^{x-5} + 2^{x-5} \cdot 2^5 = 2^{x-5}(1 + 32) = 33 \cdot 2^{x-5}$ 3. Equation: $33 \cdot 2^{x-5} = 2^6$ 4. Divide both sides by 33: $2^{x-5} = \frac{2^6}{33}$ 5. Take logarithm base 2: $x - 5 = 6 - \log_2(33)$ 6. Calculate approximate value: $x \approx 1.5$ Answer: A) 1.5 **21.** Given $2^x \cdot x^2 - 2x^2 + 2 - 2^x = 0$: 1. Group terms: $(2^x \cdot x^2 - 2x^2) + (2 - 2^x) = 0$ 2. Factor: $x^2(2^x - 2) + (2 - 2^x) = 0$ 3. Rewrite: $(2^x - 2)(x^2 - 1) = 0$ 4. Solve each factor: - $2^x - 2 = 0 \Rightarrow 2^x = 2 \Rightarrow x = 1$ - $x^2 - 1 = 0 \Rightarrow x = \pm 1$ 5. Roots: $x = 1, 1, -1$ 6. Product of roots: $1 \times 1 \times (-1) = -1$ Answer: B) -1 **22.** Given $3^2 \cdot 3^4 \cdot 3^6 \cdots 3^{2n} = (\frac{1}{81})^{-5}$: 1. Left side is product of powers: $3^{2 + 4 + 6 + \cdots + 2n}$ 2. Sum of arithmetic series: $2 + 4 + \cdots + 2n = 2(1 + 2 + \cdots + n) = 2 \cdot \frac{n(n+1)}{2} = n(n+1)$ 3. So left side: $3^{n(n+1)}$ 4. Right side: $(\frac{1}{81})^{-5} = 81^5 = (3^4)^5 = 3^{20}$ 5. Equate exponents: $n(n+1) = 20$ 6. Solve quadratic: $n^2 + n - 20 = 0$ 7. Roots: $n = \frac{-1 \pm \sqrt{1 + 80}}{2} = \frac{-1 \pm 9}{2}$ 8. Positive root: $n = 4$ Answer: A) 4 **23.** Given $5^2 \cdot 5^4 \cdot 5^6 \cdots 5^{2n} = 0.04^{-28}$: 1. Left side: $5^{2 + 4 + 6 + \cdots + 2n} = 5^{n(n+1)}$ 2. Right side: $0.04 = \frac{4}{100} = \frac{1}{25} = 5^{-2}$ 3. So $0.04^{-28} = (5^{-2})^{-28} = 5^{56}$ 4. Equate exponents: $n(n+1) = 56$ 5. Solve quadratic: $n^2 + n - 56 = 0$ 6. Roots: $n = \frac{-1 \pm \sqrt{1 + 224}}{2} = \frac{-1 \pm 15}{2}$ 7. Positive root: $n = 7$ Answer: D) 7 **24.** Given $2x^2 - 6x - 5 = 16\sqrt{2}$: 1. Rewrite: $2x^2 - 6x - 5 - 16\sqrt{2} = 0$ 2. Divide entire equation by 2: $x^2 - 3x - \frac{5}{2} - 8\sqrt{2} = 0$ 3. Sum of roots: $S = 3$ 4. Product of roots: $P = -\frac{5}{2} - 8\sqrt{2}$ 5. Calculate approximate value: $P \approx -7$ Answer: A) -7