Combine and simplify these radicals.8⋅20\sqrt{8}\cdot\sqrt{20}8​⋅20​.

To combine and simplify the given radicals, follow these steps:

Step 1: Define the Problem

Identify the expression that needs simplification: $8 \cdot \sqrt{8} \cdot \sqrt{20} \cdot 8 \cdot 20$.

Step 2: Break Down the Problem

Divide the expression into manageable parts:

• Simplify the radicals: $\sqrt{8}$ and $\sqrt{20}$.
• Simplify the multiplication of constants: $8 \cdot 8 \cdot 20$.

Step 3: Apply Relevant Concepts

1. Simplify the radicals:

   • $\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}$
   • $\sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5}$

2. Multiply the simplified radicals:

   • $\sqrt{8} \cdot \sqrt{20} = (2\sqrt{2}) \cdot (2\sqrt{5}) = 4\sqrt{10}$

3. Simplify the multiplication of constants:

   • $8 \cdot 8 \cdot 20 = 1280$

Step 4: Analysis, Verify and Summarize

Combine the simplified parts:

• Multiply the constant with the simplified radical:
  • $1280 \cdot 4\sqrt{10} = 5120\sqrt{10}$

Final Answer

The simplified form of the expression is $5120\sqrt{10}$.