11. \( \frac{\sqrt{5 n}}{2 \sqrt{15 n^{4}}} \)

To simplify the expression \( \frac{\sqrt{5 n}}{2 \sqrt{15 n^{4}}} \), start by simplifying the square roots in the denominator. We can rewrite it as: \[ \frac{\sqrt{5 n}}{2 \sqrt{15 n^{4}}} = \frac{\sqrt{5 n}}{2 \sqrt{15} \cdot \sqrt{n^{4}}} \] Since \(\sqrt{n^4} = n^2\), the expression becomes: \[ \frac{\sqrt{5 n}}{2 \sqrt{15} n^2} \] Next, since \(\sqrt{5 n} = \sqrt{5} \cdot \sqrt{n}\), we can rewrite the expression as: \[ \frac{\sqrt{5} \cdot \sqrt{n}}{2 \cdot \sqrt{15} \cdot n^2} \] This simplifies further to: \[ \frac{\sqrt{5}}{2\sqrt{15}} \cdot \frac{\sqrt{n}}{n^2} = \frac{\sqrt{5}}{2\sqrt{15}} \cdot \frac{1}{n^{3/2}} = \frac{\sqrt{5}}{2\sqrt{15}n^{3/2}} \] Now, we can simplify \(\frac{\sqrt{5}}{2\sqrt{15}}\) by rationalizing the denominator. Multiplying the numerator and denominator by \(\sqrt{15}\): \[ = \frac{\sqrt{5} \cdot \sqrt{15}}{2 \cdot 15} = \frac{\sqrt{75}}{30} = \frac{5\sqrt{3}}{30} = \frac{\sqrt{3}}{6} \] Thus, the final simplified expression is: \[ \frac{\sqrt{3}}{6n^{3/2}} \]