To simplify the expression \( 3\sqrt{54} \), we start by simplifying the square root of \( 54 \).
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Factor \( 54 \): \[ 54 = 9 \times 6 = 3^2 \times 6 \]
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Now we can simplify \( \sqrt{54} \): \[ \sqrt{54} = \sqrt{9 \times 6} = \sqrt{9} \cdot \sqrt{6} = 3\sqrt{6} \]
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Replacing \( \sqrt{54} \) in the original expression: \[ 3\sqrt{54} = 3(3\sqrt{6}) = 9\sqrt{6} \]
So the simplified form of \( 3\sqrt{54} \) is \( 9\sqrt{6} \).
Now, regarding your question about the coefficients outside the cube root, it seems there might be a slight confusion as you've mentioned cube root instead of square root. In this case, we’re only dealing with square roots.
In the expression \( 3\sqrt{54} \), the number 3 is a coefficient that multiplies the square root of 54. When we simplify the square root, we multiply the coefficients accordingly.
In short, \( 3\sqrt{54} \) simplifies to \( 9\sqrt{6} \), not involving any cube roots in this process.