1. **State the problem:** Simplify the expression $8\sqrt{17^7}$. 2. **Recall the rule:** The square root of a power can be written as $\sqrt{a^b} = a^{\frac{b}{2}}$. 3. **Apply the rule:** $$8\sqrt{17^7} = 8 \times 17^{\frac{7}{2}}$$ 4. **Rewrite the exponent:** $$17^{\frac{7}{2}} = 17^{3 + \frac{1}{2}} = 17^3 \times 17^{\frac{1}{2}} = 17^3 \times \sqrt{17}$$ 5. **Substitute back:** $$8 \times 17^3 \times \sqrt{17}$$ 6. **Calculate $17^3$:** $$17^3 = 17 \times 17 \times 17 = 4913$$ 7. **Final simplified form:** $$8 \times 4913 \times \sqrt{17} = 39304 \sqrt{17}$$ **Answer:** $39304 \sqrt{17}$