1. **State the problem:** Simplify the expression $$\frac{(m^{-1})^{-1} \cdot (m^2)^{-5}}{m^{-1} \cdot m^5}$$. 2. **Apply the power of a power rule:** $$(a^b)^c = a^{bc}$$. 3. Simplify each term: - $$(m^{-1})^{-1} = m^{-1 \times -1} = m^1 = m$$ - $$(m^2)^{-5} = m^{2 \times -5} = m^{-10}$$ 4. Substitute back: $$\frac{m \cdot m^{-10}}{m^{-1} \cdot m^5}$$ 5. Use the product of powers rule: $$a^b \cdot a^c = a^{b+c}$$. 6. Simplify numerator: $$m^{1 + (-10)} = m^{-9}$$ 7. Simplify denominator: $$m^{-1 + 5} = m^4$$ 8. Rewrite the expression: $$\frac{m^{-9}}{m^4}$$ 9. Use the quotient of powers rule: $$\frac{a^b}{a^c} = a^{b-c}$$. 10. Simplify: $$m^{-9 - 4} = m^{-13}$$ **Final answer:** $$m^{-13}$$