Question: Which expression is equivalent to $$\frac{6^{-3}}{(6^{-6})^2}$$?
1. **State the problem:** We want to simplify the expression $$\frac{6^{-3}}{(6^{-6})^2}$$ and find which equivalent expression it matches.
2. **Recall exponent rules:**
- When raising a power to another power, multiply the exponents: $$(a^m)^n = a^{m \times n}$$
- When dividing powers with the same base, subtract the exponents: $$\frac{a^m}{a^n} = a^{m-n}$$
3. **Simplify the denominator:**
$$(6^{-6})^2 = 6^{-6 \times 2} = 6^{-12}$$
4. **Rewrite the original expression:**
$$\frac{6^{-3}}{6^{-12}}$$
5. **Apply division rule:**
$$6^{-3 - (-12)} = 6^{-3 + 12} = 6^9$$
6. **Final answer:** The expression is equivalent to $$6^9$$.