Question 8

First, simplify the logarithms: Given $a = 3\sqrt{2} \implies a^2 = (3\sqrt{2})^2 = 18$. So, $\log_a (18)^{5/4} = \frac{5}{4} \log_a (a^2) = \frac{5}{4} imes 2 = \frac{5}{2} = 2.5$. The first equation is $3x + 2y = 2.5$ ... (1)

Now for $b$: $b = \frac{1}{5^{1/6} \cdot 6^{1/2}} \implies b = (5^{1/6} \cdot 6^{3/6})^{-1} = (5 \cdot 6^3)^{-1/6} = (5 \cdot 216)^{-1/6} = (1080)^{-1/6}$. Thus, $b^6 = 1080^{-1}$ or $1080 = b^{-6}$. $\log_b (\sqrt{1080}) = \log_b (1080)^{1/2} = \log_b (b^{-6})^{1/2} = \log_b (b^{-3}) = -3$. The second equation is $2x - y = -3$ ... (2)

From (2), $y = 2x + 3$. Substitute in (1): $3x + 2(2x + 3) = 2.5 \implies 3x + 4x + 6 = 2.5 \implies 7x = -3.5 \implies x = -0.5$. Then $y = 2(-0.5) + 3 = 2$. We need to find $4x + 5y = 4(-0.5) + 5(2) = -2 + 10 = 8$.