1. **State the problem:** Simplify the expression $$(17^6)^5 \cdot (17^7)^{-8}$$ giving the answer with a positive exponent. 2. **Recall the exponent rules:** - Power of a power: $$(a^m)^n = a^{m \cdot n}$$ - Product of powers with the same base: $$a^m \cdot a^n = a^{m+n}$$ - Negative exponent: $$a^{-m} = \frac{1}{a^m}$$ 3. **Apply the power of a power rule:** $$(17^6)^5 = 17^{6 \cdot 5} = 17^{30}$$ $$(17^7)^{-8} = 17^{7 \cdot (-8)} = 17^{-56}$$ 4. **Rewrite the original expression:** $$17^{30} \cdot 17^{-56}$$ 5. **Apply the product of powers rule:** $$17^{30 + (-56)} = 17^{30 - 56} = 17^{-26}$$ 6. **Rewrite with a positive exponent:** $$17^{-26} = \frac{1}{17^{26}}$$ **Final answer:** $$\boxed{\frac{1}{17^{26}}}$$