To combine the radicals in the expression \( 4\sqrt{396} + \sqrt{44} - 2\sqrt{275} \), we will simplify each radical first:
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Simplifying \( \sqrt{396} \):
To simplify \( \sqrt{396} \):
- Break down \( 396 \) into its prime factors: \[ 396 = 4 \times 99 = 4 \times 9 \times 11 = 36 \times 11. \]
- Thus, \( \sqrt{396} = \sqrt{36 \times 11} = \sqrt{36} \times \sqrt{11} = 6\sqrt{11} \).
- Therefore, \( 4\sqrt{396} = 4 \times 6\sqrt{11} = 24\sqrt{11} \).
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Simplifying \( \sqrt{44} \):
To simplify \( \sqrt{44} \):
- Break down \( 44 \): \[ 44 = 4 \times 11. \]
- Therefore, \( \sqrt{44} = \sqrt{4 \times 11} = \sqrt{4} \times \sqrt{11} = 2\sqrt{11} \).
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Simplifying \( \sqrt{275} \):
To simplify \( \sqrt{275} \):
- Break down \( 275 \): \[ 275 = 25 \times 11. \]
- Therefore, \( \sqrt{275} = \sqrt{25 \times 11} = \sqrt{25} \times \sqrt{11} = 5\sqrt{11} \).
- Thus, \( -2\sqrt{275} = -2 \times 5\sqrt{11} = -10\sqrt{11} \).
Now, we can rewrite the entire expression:
\[ 4\sqrt{396} + \sqrt{44} - 2\sqrt{275} = 24\sqrt{11} + 2\sqrt{11} - 10\sqrt{11}. \]
Next, combine the terms involving \( \sqrt{11} \):
\[ (24 + 2 - 10)\sqrt{11} = 16\sqrt{11}. \]
Thus, the final simplified form of the expression is:
\[ \boxed{16\sqrt{11}}. \]
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