Given:

$ a = 3\sqrt{2} \Rightarrow a^2 = 18 $

Simplification of $\log_a(18)^{\frac{5}{4}}$

Since $a^2 = 18$, we have:

$ \log_a(18) = \log_a(a^2) = 2 $

Therefore:

$ \log_a(18)^{\frac{5}{4}} = \frac{5}{4} \log_a(18) = \frac{5}{4} \cdot 2 = \frac{5}{2} $

So the equation becomes:

$ 3x + 2y = \frac{5}{2} $

Simplification of $\log_b(\sqrt{1080})$

Given:

$ 1080 = 2^3 \cdot 3^3 \cdot 5 = 6^3 \cdot 5 $

$ b = \frac{1}{5^{1/6} \sqrt{6}} $

$ \Rightarrow \frac{1}{b} = 5^{1/6} \sqrt{6} $

$ \Rightarrow 1080^{1/6} = 5^{1/6} \cdot 6^{1/2} = \frac{1}{b} $

Taking the square root of both sides:

$ \sqrt{1080} = \frac{1}{b^3} $

Thus:

$ \log_b(\sqrt{1080}) = \log_b\left(\frac{1}{b^3}\right) = \log_b(b^{-3}) = -3 $

So the second equation becomes:

$ 2x - y = -3 $

Solving the System of Equations

Now we have:

$ 3x + 2y = \frac{5}{2} $

$ 2x - y = -3 $

Multiply the second equation by 2:

$ 4x - 2y = -6 $

Add this to the first equation:

$ 3x + 2y + 4x - 2y = \frac{5}{2} - 6 $

$ 7x = \frac{5}{2} - 6 $

$ 7x = \frac{5}{2} - \frac{12}{2} $

$ 7x = \frac{5 - 12}{2} $

$ 7x = -\frac{7}{2} $

$ x = -\frac{1}{2} $

Substitute $x$ back into $2x - y = -3$:

$ 2\left(-\frac{1}{2}\right) - y = -3 $

$ -1 - y = -3 $

$ -y = -2 $

$ y = 2 $

Finding $4x + 5y$

$ 4x + 5y = 4 \left(-\frac{1}{2}\right) + 5 \cdot 2 $

$ = -2 + 10 $

$ = 8 $

Thus, the value of $4x + 5y$ is $\boxed{8}$.