To simplify the expression \( 2 \sqrt{12} \cdot \sqrt{27} \cdot 2 \sqrt{25} \), we can follow these steps:
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Combine the constants: \[ 2 \cdot 2 = 4. \]
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Combine the square roots: \[ \sqrt{12} \cdot \sqrt{27} \cdot \sqrt{25} = \sqrt{12 \cdot 27 \cdot 25}. \]
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Calculate the product under the square root: First, we find \( 12 \), \( 27 \), and \( 25 \) in terms of their prime factors:
- \( 12 = 2^2 \cdot 3 \)
- \( 27 = 3^3 \)
- \( 25 = 5^2 \)
Thus, we can compute the product: \[ 12 \cdot 27 \cdot 25 = (2^2 \cdot 3) \cdot (3^3) \cdot (5^2). \]
Combine these: \[ = 2^2 \cdot 3^{1+3} \cdot 5^2 = 2^2 \cdot 3^4 \cdot 5^2. \]
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Taking the square root: Using the property \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \): \[ \sqrt{12 \cdot 27 \cdot 25} = \sqrt{2^2} \cdot \sqrt{3^4} \cdot \sqrt{5^2} = 2 \cdot 3^2 \cdot 5 = 2 \cdot 9 \cdot 5. \]
Now, calculating \( 2 \cdot 9 \cdot 5 \): \[ 2 \cdot 9 = 18, \] \[ 18 \cdot 5 = 90. \]
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Final product: Therefore, \[ \sqrt{12 \cdot 27 \cdot 25} = 90. \] Now, putting it all together: \[ 4 \cdot 90 = 360. \]
Thus, the final product of the expression \( 2 \sqrt{12} \cdot \sqrt{27} \cdot 2 \sqrt{25} \) is: \[ \boxed{360}. \]