To combine the radicals \( 4\sqrt{396} + \sqrt{44} - 2\sqrt{275} \), we need to simplify each radical.
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Simplifying \( \sqrt{396} \): \[ 396 = 4 \times 99 = 4 \times 9 \times 11 \] \[ \sqrt{396} = \sqrt{4 \times 99} = \sqrt{4} \times \sqrt{99} = 2\sqrt{99} \] Now, we simplify \( \sqrt{99} \): \[ \sqrt{99} = \sqrt{9 \times 11} = \sqrt{9} \times \sqrt{11} = 3\sqrt{11} \] Therefore, \[ \sqrt{396} = 2\sqrt{99} = 2 \times 3\sqrt{11} = 6\sqrt{11} \]
Now, substituting back into the expression: \[ 4\sqrt{396} = 4 \times 6\sqrt{11} = 24\sqrt{11} \]
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Simplifying \( \sqrt{44} \): \[ 44 = 4 \times 11 \] \[ \sqrt{44} = \sqrt{4 \times 11} = \sqrt{4} \times \sqrt{11} = 2\sqrt{11} \]
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Simplifying \( \sqrt{275} \): \[ 275 = 25 \times 11 \] \[ \sqrt{275} = \sqrt{25 \times 11} = \sqrt{25} \times \sqrt{11} = 5\sqrt{11} \] Thus, \[ -2\sqrt{275} = -2 \times 5\sqrt{11} = -10\sqrt{11} \]
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Combining all the simplified terms: Now we replace back into the original expression: \[ 4\sqrt{396} + \sqrt{44} - 2\sqrt{275} = 24\sqrt{11} + 2\sqrt{11} - 10\sqrt{11} \] Combining these: \[ (24 + 2 - 10)\sqrt{11} = 16\sqrt{11} \]
Thus, the simplified form of the expression \( 4\sqrt{396} + \sqrt{44} - 2\sqrt{275} \) is:
\[ \boxed{16\sqrt{11}} \]