Trignometry (3)

$\begin{aligned} \frac{\sin^4 x}{2} + \frac{\cos^4 x}{3} & = \frac{1}{5} \\ 15\sin^4 x + 10\cos^4 x & = 6 \\ 15(1-\cos^2 x)^2 + 10\cos^4 x & = 6 \\ 15(1-2\cos^2 x + \cos^4 x) + 10\cos^4 x & = 6 \\ 25 \cos^4 x - 30 \cos^2 x + 9 & = 0 \\ (5 \cos^2 x - 3)^2 & = 0 \\ \Rightarrow \cos^2 x & = \frac{3}{5} \\ \Rightarrow \sin^2 x & = 1 - \cos^2 x = 1 - \frac{3}{5} = \frac{2}{5} \\ & \\ \Rightarrow 81000 \left(\frac{\sin^8 x}{8} + \frac{\cos^8 x}{27} \right) & = 81000 \left(\frac{1}{8} \left( \frac{2}{5}\right)^4 + \frac{1}{27} \left( \frac{3}{5}\right)^4 \right) \\ & = 81000 \left(\frac{2}{5^4} + \frac{3}{5^4} \right) \\ & = \frac{81000}{5^3} \\ & = \boxed{648} \end{aligned}$