To simplify the radical expression \( \sqrt{128x^4y^9} \), we can break it down as follows:
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Factor the radicand:
- \( 128 = 64 \cdot 2 = 8^2 \cdot 2 = 4^2 \cdot 8 = 2^7 \)
- For \( x^4 \), it is already simplified.
- For \( y^9 \), we can write it as \( (y^4)^2 \cdot y \) (since 9 is an even number).
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Rewrite the expression under the square root: \[ \sqrt{128x^4y^9} = \sqrt{(8^2 \cdot 2) \cdot x^4 \cdot (y^4)^2 \cdot y} \]
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Separate the perfect squares from the rest: \[ = \sqrt{8^2} \cdot \sqrt{x^4} \cdot \sqrt{(y^4)^2} \cdot \sqrt{2} \cdot \sqrt{y} \]
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Simplify the square roots:
- \( \sqrt{8^2} = 8 \)
- \( \sqrt{x^4} = x^2 \)
- \( \sqrt{(y^4)^2} = y^4 \)
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Combine everything together: \[ = 8x^2y^4\sqrt{2y} \]
Now, we can compare to the responses provided. The simplified result \( 8x^2y^4\sqrt{2y} \) matches with the option that represents this expression.
Therefore, the final simplified form of \( \sqrt{128x^4y^9} \) is: \[ 8x^2y^4\sqrt{2y} \]
Based on the provided options, none precisely matches this exact expression, but we can see that it simplifies consistently. If you have a specific option in mind and want a detailed check against it, feel free to point it out!