1. **Problem statement:** Simplify the expression $$\frac{a^{-2} + a^{-2} + a^{-2}}{a^{-5} + a^{-5} + a^{-5}}$$. 2. **Recall the rules:** - When adding like terms, add their coefficients. - Negative exponents mean reciprocal powers: $$a^{-n} = \frac{1}{a^n}$$. - When dividing powers with the same base, subtract exponents: $$\frac{a^m}{a^n} = a^{m-n}$$. 3. **Simplify numerator:** $$a^{-2} + a^{-2} + a^{-2} = 3a^{-2}$$ 4. **Simplify denominator:** $$a^{-5} + a^{-5} + a^{-5} = 3a^{-5}$$ 5. **Rewrite the fraction:** $$\frac{3a^{-2}}{3a^{-5}}$$ 6. **Cancel common factor 3:** $$\frac{a^{-2}}{a^{-5}}$$ 7. **Apply division rule for exponents:** $$a^{-2 - (-5)} = a^{-2 + 5} = a^{3}$$ **Final answer:** $$a^{3}$$