To simplify the radical \(\sqrt{128x^4y^9}\), we can break it down as follows:
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Factor the number and use properties of square roots:
- \(128\) can be factored into \(64 \cdot 2\) where \(64\) is a perfect square (\(8^2\)).
- The \(x^4\) can be written as \((x^2)^2\), which is also a perfect square.
- The \(y^9\) can be written as \((y^4)^2 \cdot y\), since \(y^8\) is a perfect square.
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Apply the square root property: \[ \sqrt{128x^4y^9} = \sqrt{64 \cdot 2} \cdot \sqrt{x^4} \cdot \sqrt{y^9} \]
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Calculate the square roots:
- \(\sqrt{64} = 8\)
- \(\sqrt{x^4} = x^2\)
- \(\sqrt{y^9} = y^4 \cdot \sqrt{y}\) (because \(\sqrt{y^8} = y^4\))
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Combine the results: \[ \sqrt{128x^4y^9} = 8 x^2 y^4 \sqrt{2y} \]
Thus, the simplified form is: \[ 8x^2y^4\sqrt{2y} \]
So, the correct response is: 8 x squared y to the 4th power square root of 2 y.
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