To find the sum of the expression \( 2\sqrt{108} + \sqrt{48} + 5\sqrt{27} \), we will simplify each term individually and then combine them.
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Simplifying \( \sqrt{108} \): \[ \sqrt{108} = \sqrt{36 \times 3} = \sqrt{36} \cdot \sqrt{3} = 6\sqrt{3} \] Therefore, \[ 2\sqrt{108} = 2 \cdot 6\sqrt{3} = 12\sqrt{3} \]
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Simplifying \( \sqrt{48} \): \[ \sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \cdot \sqrt{3} = 4\sqrt{3} \]
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Simplifying \( 5\sqrt{27} \): \[ \sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3} \] Therefore, \[ 5\sqrt{27} = 5 \cdot 3\sqrt{3} = 15\sqrt{3} \]
Now we can combine all the simplified terms: \[ 2\sqrt{108} + \sqrt{48} + 5\sqrt{27} = 12\sqrt{3} + 4\sqrt{3} + 15\sqrt{3} \] \[ = (12 + 4 + 15)\sqrt{3} = 31\sqrt{3} \]
Thus, the sum of the expression is: \[ \boxed{31\sqrt{3}} \]
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