1. **State the problem:** Simplify the expression $\left(5\sqrt{8}\right)^{\frac{5}{2}} \times 16^{-\frac{3}{2}}$. 2. **Rewrite the terms:** Recall that $\sqrt{8} = 8^{\frac{1}{2}}$ and $16 = 2^4$. 3. **Express inside the first term:** $$5\sqrt{8} = 5 \times 8^{\frac{1}{2}} = 5 \times \left(2^3\right)^{\frac{1}{2}} = 5 \times 2^{\frac{3}{2}}$$ 4. **Rewrite the first term with powers:** $$\left(5 \times 2^{\frac{3}{2}}\right)^{\frac{5}{2}} = 5^{\frac{5}{2}} \times \left(2^{\frac{3}{2}}\right)^{\frac{5}{2}} = 5^{\frac{5}{2}} \times 2^{\frac{3}{2} \times \frac{5}{2}} = 5^{\frac{5}{2}} \times 2^{\frac{15}{4}}$$ 5. **Rewrite the second term:** $$16^{-\frac{3}{2}} = \left(2^4\right)^{-\frac{3}{2}} = 2^{4 \times -\frac{3}{2}} = 2^{-6}$$ 6. **Combine the terms:** $$5^{\frac{5}{2}} \times 2^{\frac{15}{4}} \times 2^{-6} = 5^{\frac{5}{2}} \times 2^{\frac{15}{4} - 6}$$ 7. **Simplify the exponent of 2:** $$\frac{15}{4} - 6 = \frac{15}{4} - \frac{24}{4} = -\frac{9}{4}$$ 8. **Final expression:** $$5^{\frac{5}{2}} \times 2^{-\frac{9}{4}}$$ 9. **Optional: express in radical form:** $$5^{\frac{5}{2}} = \left(5^{\frac{1}{2}}\right)^5 = \left(\sqrt{5}\right)^5$$ $$2^{-\frac{9}{4}} = \frac{1}{2^{\frac{9}{4}}} = \frac{1}{\left(2^{\frac{1}{4}}\right)^9} = \frac{1}{\left(\sqrt[4]{2}\right)^9}$$ **Answer:** $$\boxed{5^{\frac{5}{2}} \times 2^{-\frac{9}{4}}}$$