1. **State the problem:** Simplify the expression $5\sqrt{7} + 3\sqrt{40}$. 2. **Recall the rule:** To simplify expressions with square roots, factor the radicand (number inside the root) to extract perfect squares. 3. **Simplify $\sqrt{40}$:** $$\sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10}$$ 4. **Substitute back:** $$5\sqrt{7} + 3\sqrt{40} = 5\sqrt{7} + 3 \times 2\sqrt{10} = 5\sqrt{7} + 6\sqrt{10}$$ 5. **Final expression:** Since $\sqrt{7}$ and $\sqrt{10}$ are different surds, the expression cannot be simplified further. **Answer:** $5\sqrt{7} + 6\sqrt{10}$