To combine the radicals \( 4\sqrt{396} + \sqrt{44} - 2\sqrt{275} \), we first simplify each radical term individually.
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Simplifying \( \sqrt{396} \): \[ 396 = 4 \times 99 = 4 \times 9 \times 11 = 36 \times 11 \] Thus, \[ \sqrt{396} = \sqrt{4 \times 99} = \sqrt{4} \times \sqrt{99} = 2\sqrt{99} \] Now we can compute \( 4\sqrt{396} \): \[ 4\sqrt{396} = 4 \times 2\sqrt{99} = 8\sqrt{99} \]
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Simplifying \( \sqrt{44} \): \[ 44 = 4 \times 11 \] Thus, \[ \sqrt{44} = \sqrt{4 \times 11} = \sqrt{4} \times \sqrt{11} = 2\sqrt{11} \]
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Simplifying \( \sqrt{275} \): \[ 275 = 25 \times 11 = 5^2 \times 11 \] Thus, \[ \sqrt{275} = \sqrt{25 \times 11} = \sqrt{25} \times \sqrt{11} = 5\sqrt{11} \] Now we can compute \( 2\sqrt{275} \): \[ 2\sqrt{275} = 2 \times 5\sqrt{11} = 10\sqrt{11} \]
Now we substitute back into the original expression:
\[ 4\sqrt{396} + \sqrt{44} - 2\sqrt{275} = 8\sqrt{99} + 2\sqrt{11} - 10\sqrt{11} \]
Next, we combine the \(\sqrt{11}\) terms:
\[ 8\sqrt{99} + (2\sqrt{11} - 10\sqrt{11}) = 8\sqrt{99} - 8\sqrt{11} \]
Finally, we factor out the common term:
\[ 8\sqrt{99} - 8\sqrt{11} = 8(\sqrt{99} - \sqrt{11}) \]
Thus, the final result in simplest radical form is:
\[ \boxed{8(\sqrt{99} - \sqrt{11})} \]
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