To simplify the expression \(\sqrt{72x^3z^3}\), we can break it down as follows:
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Factor the number: Start by factoring \(72\): \[ 72 = 36 \times 2 = 6^2 \times 2 \] So, we have: \[ \sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2} \]
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Simplify the variable parts: Now let's simplify \(\sqrt{x^3}\) and \(\sqrt{z^3}\): \[ \sqrt{x^3} = \sqrt{x^2 \cdot x} = \sqrt{x^2} \cdot \sqrt{x} = x\sqrt{x} \] Similarly for \(z\): \[ \sqrt{z^3} = \sqrt{z^2 \cdot z} = \sqrt{z^2} \cdot \sqrt{z} = z\sqrt{z} \]
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Combine everything: Now we can put all these pieces together: \[ \sqrt{72x^3z^3} = \sqrt{72} \cdot \sqrt{x^3} \cdot \sqrt{z^3} = (6\sqrt{2}) \cdot (x\sqrt{x}) \cdot (z\sqrt{z}) \]
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Final simplification: Combine all the factors: \[ = 6xyz \sqrt{2xz} \]
So, the simplified form of \(\sqrt{72x^3z^3}\) is: \[ \boxed{6xyz \sqrt{2xz}} \]
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