1. **State the problem:** We need to find the volume of a rectangular prism with side lengths $\frac{3}{4}$ cm, $\frac{1}{5}$ cm, and $\frac{5}{7}$ cm. 2. **Formula for volume of a rectangular prism:** $$\text{Volume} = \text{length} \times \text{width} \times \text{height}$$ 3. **Substitute the given values:** $$V = \frac{3}{4} \times \frac{1}{5} \times \frac{5}{7}$$ 4. **Multiply the fractions:** $$V = \frac{3 \times 1 \times 5}{4 \times 5 \times 7} = \frac{15}{140}$$ 5. **Simplify the fraction:** Cancel the common factor 5: $$V = \frac{\cancel{15}^{3}}{\cancel{140}^{28}} = \frac{3}{28}$$ 6. **Final answer:** The volume of the rectangular prism is $$\boxed{\frac{3}{28} \text{ cm}^3}$$