To simplify the expression \( 2\sqrt{12} \cdot \sqrt{27} \cdot 2\sqrt{25} \), we can break it down step by step.
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First, let's rewrite each square root in a simpler form:
- \( \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3} \)
- \( \sqrt{27} = \sqrt{9 \cdot 3} = 3\sqrt{3} \)
- \( \sqrt{25} = 5 \)
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Now, substituting these back into the expression yields: \[ 2\sqrt{12} \cdot \sqrt{27} \cdot 2\sqrt{25} = 2(2\sqrt{3}) \cdot (3\sqrt{3}) \cdot 2(5) \]
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This simplifies to: \[ = 2 \cdot 2 \cdot 2 \cdot 5 \cdot 3 \cdot \sqrt{3} \cdot \sqrt{3} \] \[ = 8 \cdot 5 \cdot 3 \cdot \sqrt{3} \cdot \sqrt{3} \]
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Since \( \sqrt{3} \cdot \sqrt{3} = 3 \), we have: \[ = 8 \cdot 5 \cdot 3 \cdot 3 \] \[ = 8 \cdot 5 \cdot 9 \]
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Now we can simplify further:
- \( 8 \cdot 5 = 40 \)
- \( 40 \cdot 9 = 360 \)
Thus, the product of the expression in simplest radical form is: \[ \boxed{360} \]