jmc

Combining the expressions on the left-hand side, we obtain $$\begin{array}{rl}\sqrt{8x}\cdot \sqrt{10x}\cdot \sqrt{3x}\cdot \sqrt{15x}& =15\\ \sqrt{3600x^4}& =15\\ 60x^2& =15\\ x^2& =\frac{15}{60}=\frac{1}{4}.\end{array}$$Since $x$ must be positive, the only solution is $x=\sqrt{\frac{1}{4}}=\boxed{\frac{1}{2}}$.