1. **State the problem:** Simplify the expression $\frac{3\sqrt{8}}{8\sqrt{5}}$. 2. **Recall the rules:** - Simplify square roots by factoring out perfect squares. - When dividing radicals, combine under a single radical if possible. - Simplify fractions by canceling common factors. 3. **Simplify the square roots:** $$\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$$ 4. **Substitute back:** $$\frac{3 \times 2\sqrt{2}}{8 \sqrt{5}} = \frac{6\sqrt{2}}{8\sqrt{5}}$$ 5. **Simplify the fraction:** $$\frac{6\sqrt{2}}{8\sqrt{5}} = \frac{\cancel{6}^{3}\sqrt{2}}{\cancel{8}^{4}\sqrt{5}} = \frac{3\sqrt{2}}{4\sqrt{5}}$$ 6. **Rationalize the denominator:** Multiply numerator and denominator by $\sqrt{5}$: $$\frac{3\sqrt{2}}{4\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{3\sqrt{2} \times \sqrt{5}}{4 \times 5} = \frac{3\sqrt{10}}{20}$$ 7. **Final answer:** $$\boxed{\frac{3\sqrt{10}}{20}}$$