To simplify the expression \(\sqrt{f^{14}g^9h^{11}}\), we can use the property that \(\sqrt{a^b} = a^{b/2}\). Let's apply this property to each term inside the square root:
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For \(f^{14}\): \[ \sqrt{f^{14}} = f^{14/2} = f^7 \]
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For \(g^{9}\): \[ \sqrt{g^{9}} = g^{9/2} = g^{4.5} = g^4 \cdot g^{1/2} \]
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For \(h^{11}\): \[ \sqrt{h^{11}} = h^{11/2} = h^5 \cdot h^{1/2} \]
Now, putting it all together: \[ \sqrt{f^{14}g^9h^{11}} = f^7 \cdot g^4 \cdot g^{1/2} \cdot h^5 \cdot h^{1/2} \]
This simplifies to: \[ f^7 g^4 h^5 \sqrt{gh} \]
Thus, the simplified expression is: \[ f^7 g^4 h^5 \sqrt{gh} \]