To simplify \( 3 \sqrt{-375} \), we first recognize that the presence of a negative under the square root indicates that we will involve imaginary numbers.
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Start by factoring out the negative sign: \[ 3 \sqrt{-375} = 3 \sqrt{-1 \cdot 375} = 3 \sqrt{-1} \cdot \sqrt{375} \]
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Since \( \sqrt{-1} = i \) (where \( i \) is the imaginary unit), we have: \[ 3 \sqrt{-375} = 3i \sqrt{375} \]
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Next, simplify \( \sqrt{375} \). We can factor 375 as follows: \[ 375 = 25 \cdot 15 = 25 \cdot 3 \cdot 5 \] Therefore, we can simplify \( \sqrt{375} \): \[ \sqrt{375} = \sqrt{25 \cdot 15} = \sqrt{25} \cdot \sqrt{15} = 5\sqrt{15} \]
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Now substitute this back into our previous equation: \[ 3i \sqrt{375} = 3i (5 \sqrt{15}) = 15i \sqrt{15} \]
Thus, the simplified form of \( 3 \sqrt{-375} \) is: \[ \boxed{15i \sqrt{15}} \]