To simplify the expression \(\sqrt{32x^6}\), we can break it down into smaller parts.
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Start by factoring \(32\): \[ 32 = 16 \cdot 2 = 4^2 \cdot 2 \]
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For \(x^6\), we know that: \[ x^6 = (x^3)^2 \]
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Now we can rewrite the expression: \[ \sqrt{32x^6} = \sqrt{32} \cdot \sqrt{x^6} \]
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Now, evaluate \(\sqrt{32}\) and \(\sqrt{x^6}\): \[ \sqrt{32} = \sqrt{16 \cdot 2} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2} \] \[ \sqrt{x^6} = x^{6/2} = x^3 \]
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Putting it all together: \[ \sqrt{32x^6} = 4\sqrt{2} \cdot x^3 = 4x^3\sqrt{2} \]
Therefore, the expression equivalent to \(\sqrt{32x^6}\) is:
A. \(4x^3\sqrt{2}\).