1. **State the problem:** Calculate $3^4 \times 3^7$. 2. **Formula used:** When multiplying powers with the same base, add the exponents: $$a^m \times a^n = a^{m+n}$$ 3. **Apply the rule:** Here, the base is 3, so: $$3^4 \times 3^7 = 3^{4+7}$$ 4. **Simplify the exponent:** $$3^{11}$$ 5. **Calculate the value:** $$3^{11} = 177147$$ **Final answer:** $177147$