To simplify the expression \(9 \sqrt{24} - 5 \sqrt{6} + 3 \sqrt{54}\), we can start by simplifying each radical.
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Simplifying \(\sqrt{24}\): \[ \sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \cdot \sqrt{6} = 2\sqrt{6} \] Thus, \(9 \sqrt{24} = 9 \cdot 2 \sqrt{6} = 18\sqrt{6}\).
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Simplifying \(\sqrt{54}\): \[ \sqrt{54} = \sqrt{9 \times 6} = \sqrt{9} \cdot \sqrt{6} = 3\sqrt{6} \] Thus, \(3 \sqrt{54} = 3 \cdot 3 \sqrt{6} = 9\sqrt{6}\).
Now substitute these simplified forms back into the expression: \[ 18 \sqrt{6} - 5 \sqrt{6} + 9 \sqrt{6} \]
Next, combine the like terms: \[ (18 - 5 + 9) \sqrt{6} \] Calculating the coefficients: \[ 18 - 5 = 13 \] \[ 13 + 9 = 22 \]
Thus, the expression simplifies to: \[ 22 \sqrt{6} \]
So the final answer is: \[ \boxed{22 \sqrt{6}} \]