To simplify \(\sqrt{72z^5}\), we start by breaking down the expression under the square root into its factors.
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Factor the number 72: \[ 72 = 36 \times 2 = 6^2 \times 2 \] Thus, we can rewrite \(72\) as \(36 \cdot 2\).
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Break down \(z^5\): \[ z^5 = z^4 \cdot z = (z^2)^2 \cdot z \] Here, we have taken out a perfect square \(z^4\) since \((z^2)^2\) is a perfect square.
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Combine both factorizations: \[ 72z^5 = 36 \cdot 2 \cdot z^4 \cdot z = 36z^4 \cdot 2z \]
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Now apply the square root: \[ \sqrt{72z^5} = \sqrt{36z^4 \cdot 2z} \]
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Using the property of square roots \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\), we can separate the factors: \[ \sqrt{72z^5} = \sqrt{36} \cdot \sqrt{z^4} \cdot \sqrt{2z} \]
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Calculating the square roots of the perfect squares: \[ \sqrt{36} = 6 \quad \text{and} \quad \sqrt{z^4} = z^2 \]
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Combine the results: \[ \sqrt{72z^5} = 6 \cdot z^2 \cdot \sqrt{2z} \]
Thus, the simplified form is: \[ \sqrt{72z^5} = 6z^2\sqrt{2z} \]
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