To combine the radicals \( 4\sqrt{396} + \sqrt{44} - 2\sqrt{275} \), we first simplify each radical:
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Simplifying \( \sqrt{396} \): \[ 396 = 4 \times 99 = 4 \times 9 \times 11 = 36 \times 11 \] Therefore, \[ \sqrt{396} = \sqrt{36 \times 11} = \sqrt{36} \cdot \sqrt{11} = 6\sqrt{11} \] Thus, \[ 4\sqrt{396} = 4 \cdot 6\sqrt{11} = 24\sqrt{11} \]
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Simplifying \( \sqrt{44} \): \[ 44 = 4 \times 11 \] Therefore, \[ \sqrt{44} = \sqrt{4 \times 11} = \sqrt{4} \cdot \sqrt{11} = 2\sqrt{11} \]
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Simplifying \( \sqrt{275} \): \[ 275 = 25 \times 11 \] Therefore, \[ \sqrt{275} = \sqrt{25 \times 11} = \sqrt{25} \cdot \sqrt{11} = 5\sqrt{11} \] Thus, \[ -2\sqrt{275} = -2 \cdot 5\sqrt{11} = -10\sqrt{11} \]
Now, substitute the simplified forms back into the expression: \[ 4\sqrt{396} + \sqrt{44} - 2\sqrt{275} = 24\sqrt{11} + 2\sqrt{11} - 10\sqrt{11} \]
Combine the coefficients of \( \sqrt{11} \): \[ (24 + 2 - 10)\sqrt{11} = 16\sqrt{11} \]
Thus, the final simplified expression is: \[ \boxed{16\sqrt{11}} \]