1. **State the problem:** We need to find the area of a rectangle with side lengths $5\sqrt{7}$ cm and $2\sqrt{7}$ cm. 2. **Formula for the area of a rectangle:** $$\text{Area} = \text{length} \times \text{width}$$ 3. **Substitute the given values:** $$\text{Area} = 5\sqrt{7} \times 2\sqrt{7}$$ 4. **Multiply the coefficients and the square roots separately:** $$= (5 \times 2) \times (\sqrt{7} \times \sqrt{7})$$ 5. **Simplify the square root multiplication:** $$\sqrt{7} \times \sqrt{7} = 7$$ 6. **Calculate the product:** $$= 10 \times 7 = 70$$ 7. **Final answer:** The area of the rectangle is $70$ cm$^2$ in its simplest form.