1. **Problem Statement:** Simplify the expression $$\frac{3^5 \times 3^{-11}}{2^{-1} \times 5^0}$$. 2. **Formula and Rules:** When multiplying powers with the same base, add the exponents: $$a^m \times a^n = a^{m+n}$$. When dividing powers with the same base, subtract the exponents: $$\frac{a^m}{a^n} = a^{m-n}$$. Any number raised to the zero power is 1: $$a^0 = 1$$. 3. **Step-by-step Simplification:** - Simplify numerator: $$3^5 \times 3^{-11} = 3^{5 + (-11)} = 3^{-6}$$. - Simplify denominator: $$2^{-1} \times 5^0 = 2^{-1} \times 1 = 2^{-1}$$. - The expression becomes: $$\frac{3^{-6}}{2^{-1}}$$. - Dividing by a power is the same as multiplying by its reciprocal: $$\frac{3^{-6}}{2^{-1}} = 3^{-6} \times 2^{1} = 2 \times 3^{-6}$$. 4. **Final Answer:** $$2 \times 3^{-6} = \frac{2}{3^6} = \frac{2}{729}$$. Thus, the simplified form is $$\frac{2}{729}$$.