To simplify the expression \( 2\sqrt{12} \cdot \sqrt{27} \cdot 2\sqrt{25} \), we start by breaking down each square root and multiplying the components.
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Calculate each square root individually:
- \( \sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \)
- \( \sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3} \)
- \( \sqrt{25} = \sqrt{25} = 5 \)
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Substitute these values back into the original expression: \[ 2\sqrt{12} \cdot \sqrt{27} \cdot 2\sqrt{25} = 2(2\sqrt{3}) \cdot (3\sqrt{3}) \cdot (2 \cdot 5) \]
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Simplify the coefficients first:
- Coefficients: \( 2 \cdot 2 \cdot 2 \cdot 5 = 8 \cdot 5 = 40 \)
- Now, the square root terms: \[ \sqrt{3} \cdot \sqrt{3} = \sqrt{3^2} = 3 \] Therefore, we simplify the square root terms: \[ (2\sqrt{3}) \cdot 3\sqrt{3} = 2 \cdot 3 \cdot \sqrt{3} = 6\sqrt{3} \]
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Combine everything: Combine the coefficients with the square root term: \[ 40 \cdot 3 = 120 \] Thus the expression becomes: \[ 120\cdot\sqrt{3} \]
Finally, the product in simplified radical form is: \[ \boxed{120\sqrt{3}} \]
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