To simplify the expression \( \sqrt{20m} \cdot \sqrt{5n} \), we can apply the property of square roots which states that \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \).
Let's simplify the expression step by step:
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Combine the square roots: \[ \sqrt{20m} \cdot \sqrt{5n} = \sqrt{20m \cdot 5n} \]
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Calculate the product inside the square root: \[ 20m \cdot 5n = (20 \cdot 5)(m \cdot n) = 100mn \]
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Now, we have: \[ \sqrt{20m} \cdot \sqrt{5n} = \sqrt{100mn} \]
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Simplify the square root: \[ \sqrt{100mn} = \sqrt{100} \cdot \sqrt{mn} = 10\sqrt{mn} \]
Thus, the simplified form of the expression is: \[ 10\sqrt{mn} \]
So, the missing values in the solution are:
- The coefficient is 10.
- The variables are \( m \) and \( n \).
In alphabetical order, you would write:
- The final answer is \( 10\sqrt{mn} \).