1. **State the problem:** Calculate the value of $5^5 \times 40^7$. 2. **Recall the properties of exponents:** When multiplying powers, if the bases are the same, you add the exponents. Here, the bases are different, so we calculate each power separately and then multiply. 3. **Calculate each power:** - $5^5 = 5 \times 5 \times 5 \times 5 \times 5 = 3125$ - $40^7 = 40 \times 40 \times 40 \times 40 \times 40 \times 40 \times 40$ 4. **Express $40$ as $5 \times 8$ to simplify:** $$40^7 = (5 \times 8)^7 = 5^7 \times 8^7$$ 5. **Rewrite the original expression:** $$5^5 \times 40^7 = 5^5 \times 5^7 \times 8^7 = 5^{5+7} \times 8^7 = 5^{12} \times 8^7$$ 6. **Calculate $5^{12}$ and $8^7$ separately:** - $5^{12} = 244140625$ - $8^7 = 2097152$ 7. **Multiply the results:** $$244140625 \times 2097152 = 511620083045000$$ **Final answer:** $$5^5 \times 40^7 = 511620083045000$$