To simplify the expression \(\sqrt{108a^6}\), we start by factoring out the perfect square terms from inside the square root.
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Factor the number 108: \[ 108 = 36 \times 3 = 6^2 \times 3 \]
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Factor \(a^6\): \[ a^6 = (a^3)^2 \]
Now we can rewrite \(\sqrt{108a^6}\) by substituting these factors into the expression:
\[ \sqrt{108a^6} = \sqrt{36 \times 3 \times (a^3)^2} \]
- Apply the property of square roots: Using the property \(\sqrt{xy} = \sqrt{x} \cdot \sqrt{y}\), we can separate the square root into perfect squares and non-perfect squares:
\[ \sqrt{108a^6} = \sqrt{36} \cdot \sqrt{3} \cdot \sqrt{(a^3)^2} \]
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Calculate the square roots of the perfect squares: \[ \sqrt{36} = 6 \quad \text{and} \quad \sqrt{(a^3)^2} = a^3 \]
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Combine the results: \[ \sqrt{108a^6} = 6 \cdot a^3 \cdot \sqrt{3} \]
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Final answer: Therefore, the simplified form is:
\[ \sqrt{108a^6} = 6a^3\sqrt{3} \]
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