To simplify \(\sqrt{39y^9}\), we start by breaking it down into its components.
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Identify Perfect Squares:
- The number 39 can be analyzed, but it has no perfect square factors since its prime factorization is \(3 \times 13\).
- For \(y^9\), we can rewrite it as \(y^8 \cdot y^1 = (y^4)^2 \cdot y\), where \(y^4\) is a perfect square.
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Rewrite the Square Root: \[ \sqrt{39y^9} = \sqrt{39 \cdot (y^4)^2 \cdot y} \]
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Apply the Product Property of Square Roots: We can separate the terms under the square root: \[ \sqrt{39y^9} = \sqrt{39} \cdot \sqrt{(y^4)^2} \cdot \sqrt{y} \]
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Simplify the Perfect Square Term: The square root of a square is simply the base of that square: \[ \sqrt{(y^4)^2} = y^4 \]
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Combine the Results: Thus, we have: \[ \sqrt{39y^9} = y^4 \cdot \sqrt{39} \cdot \sqrt{y} = y^4 \sqrt{39y} \]
The final simplified expression is: \[ \sqrt{39y^9} = y^4 \sqrt{39y} \]
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