1. **State the problem:** We need to find which expressions are equivalent to $$3^{-9} \cdot 3^{-3}$$. 2. **Use the property of exponents:** When multiplying powers with the same base, add the exponents: $$3^{-9} \cdot 3^{-3} = 3^{-9 + (-3)} = 3^{-12}$$ 3. **Check each expression:** - Expression: $$\frac{1}{3^{-12}}$$ Rewrite denominator: $$\frac{1}{3^{-12}} = 3^{12}$$ This is not equal to $$3^{-12}$$. - Expression: $$\frac{12^{-12}}{4^{-12}}$$ Rewrite as: $$12^{-12} \cdot 4^{12}$$ Since bases differ, simplify by prime factorization: $$12 = 3 \cdot 2^2, \quad 4 = 2^2$$ So: $$12^{-12} = (3 \cdot 2^2)^{-12} = 3^{-12} \cdot 2^{-24}$$ $$4^{12} = (2^2)^{12} = 2^{24}$$ Multiply: $$3^{-12} \cdot 2^{-24} \cdot 2^{24} = 3^{-12} \cdot \cancel{2^{-24} \cdot 2^{24}} = 3^{-12}$$ This matches $$3^{-12}$$. - Expression: $$(3^{-3})^4$$ Use power of a power rule: $$(3^{-3})^4 = 3^{-3 \cdot 4} = 3^{-12}$$ Matches $$3^{-12}$$. - Expression: $$(3^{-4})^{-8}$$ Use power of a power rule: $$(3^{-4})^{-8} = 3^{-4 \cdot (-8)} = 3^{32}$$ Does not match $$3^{-12}$$. 4. **Summary:** The expressions equivalent to $$3^{-9} \cdot 3^{-3}$$ are: - $$\frac{12^{-12}}{4^{-12}}$$ - $$(3^{-3})^4$$ 5. **Final answer:** $$3^{-9} \cdot 3^{-3} = 3^{-12} = \frac{12^{-12}}{4^{-12}} = (3^{-3})^4$$