To combine the radicals \( 6\sqrt{117} - 3\sqrt{52} + 4\sqrt{13} \), we first simplify each radical where possible.
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Simplify \( \sqrt{117} \): \[ \sqrt{117} = \sqrt{9 \times 13} = \sqrt{9} \cdot \sqrt{13} = 3\sqrt{13} \] Thus, \( 6\sqrt{117} = 6(3\sqrt{13}) = 18\sqrt{13} \).
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Simplify \( \sqrt{52} \): \[ \sqrt{52} = \sqrt{4 \times 13} = \sqrt{4} \cdot \sqrt{13} = 2\sqrt{13} \] Thus, \( -3\sqrt{52} = -3(2\sqrt{13}) = -6\sqrt{13} \).
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Combine the terms: Now we can combine all the terms: \[ 18\sqrt{13} - 6\sqrt{13} + 4\sqrt{13} \] Combine like terms: \[ (18 - 6 + 4)\sqrt{13} = 16\sqrt{13} \]
The final answer in simplest radical form is: \[ \boxed{16\sqrt{13}} \]
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