1. The problem is to simplify $$\frac{4^{-4}}{4^{-3}}$$ and express the answer in exponential form with base 2. 2. Recall the rule for dividing powers with the same base: $$\frac{a^m}{a^n} = a^{m-n}$$. 3. Apply this rule to the given expression: $$\frac{4^{-4}}{4^{-3}} = 4^{-4 - (-3)} = 4^{-4 + 3} = 4^{-1}$$. 4. Now express 4 as a power of 2: $$4 = 2^2$$. 5. Substitute into the expression: $$4^{-1} = (2^2)^{-1}$$. 6. Use the power of a power rule: $$(a^m)^n = a^{mn}$$. 7. Simplify: $$(2^2)^{-1} = 2^{2 \times (-1)} = 2^{-2}$$. 8. Final answer: $$\boxed{2^{-2}}$$. This is the simplified form of the original expression with base 2.