1. **State the problem:** Multiply the expressions $2\sqrt{7}$ and $6\sqrt{35}$ and simplify the result. 2. **Recall the multiplication rule for radicals:** When multiplying square roots, use the property: $$\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$$ 3. **Multiply the coefficients (numbers outside the radicals):** $$2 \times 6 = 12$$ 4. **Multiply the radicals:** $$\sqrt{7} \cdot \sqrt{35} = \sqrt{7 \times 35}$$ 5. **Calculate the product inside the radical:** $$7 \times 35 = 245$$ 6. **Rewrite the expression:** $$12 \sqrt{245}$$ 7. **Simplify the radical $\sqrt{245}$:** Find the prime factors of 245: $$245 = 7 \times 35 = 7 \times 7 \times 5 = 7^2 \times 5$$ 8. **Use the property $\sqrt{a^2 \times b} = a \sqrt{b}$:** $$\sqrt{245} = \sqrt{7^2 \times 5} = 7 \sqrt{5}$$ 9. **Substitute back:** $$12 \sqrt{245} = 12 \times 7 \sqrt{5} = 84 \sqrt{5}$$ **Final answer:** $$\boxed{84 \sqrt{5}}$$