1. **State the problem:** Simplify the expression $$2^{-1} \cdot 2^{6}$$ using the properties of exponents. 2. **Recall the property of exponents:** When multiplying powers with the same base, add the exponents: $$a^{m} \cdot a^{n} = a^{m+n}$$ 3. **Apply the property:** $$2^{-1} \cdot 2^{6} = 2^{-1+6}$$ 4. **Simplify the exponent:** $$2^{5}$$ 5. **Final answer:** $$2^{5} = 32$$ So, the simplified expression is $$32$$.