1. Stating the problem: Simplify the expression $$\sqrt{10^{400} \div 10^{220} \times 10^{80}}$$. 2. Apply the division rule for exponents: $$10^{400} \div 10^{220} = 10^{400-220} = 10^{180}$$. 3. Now multiply by $$10^{80}$$: $$10^{180} \times 10^{80} = 10^{180+80} = 10^{260}$$. 4. The expression inside the square root is now $$10^{260}$$, so the problem becomes simplifying $$\sqrt{10^{260}}$$. 5. Use the property of square roots and powers: $$\sqrt{10^{260}} = 10^{260/2} = 10^{130}$$. 6. Therefore, the simplified expression is $$10^{130}$$.