To simplify the expression \(3\sqrt{5} \cdot 2\sqrt{10}\), we will follow these steps:
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Multiply the coefficients (the numbers outside the square roots): \[ 3 \cdot 2 = 6 \]
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Multiply the square roots: \[ \sqrt{5} \cdot \sqrt{10} = \sqrt{5 \cdot 10} = \sqrt{50} \]
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Combine the results from steps 1 and 2: \[ 3\sqrt{5} \cdot 2\sqrt{10} = 6 \cdot \sqrt{50} \]
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Simplify \(\sqrt{50}\):
- First, factor \(50\): \[ 50 = 25 \cdot 2 \]
- Then, take the square root: \[ \sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2} \]
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Substitute \(\sqrt{50}\) back into the expression: \[ 6 \cdot \sqrt{50} = 6 \cdot 5\sqrt{2} = 30\sqrt{2} \]
Thus, the simplified expression is: \[ \boxed{30\sqrt{2}} \]