To complete the expression \( ? \sqrt{12} + 2 \sqrt{27} = 14\sqrt{?} \), we should first simplify the radicals on the left side and then solve for the unknown.
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Simplify \(\sqrt{12}\) and \(2\sqrt{27}\): \[ \sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \] \[ 2\sqrt{27} = 2\sqrt{9 \cdot 3} = 2 \cdot 3 \cdot \sqrt{3} = 6\sqrt{3} \]
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Combine the simplified terms: \[ ? \sqrt{12} + 2 \sqrt{27} = ?(2\sqrt{3}) + 6\sqrt{3} \] This becomes: \[ (2? + 6)\sqrt{3} \]
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Equate this to the right side of the original equation: \[ (2? + 6)\sqrt{3} = 14\sqrt{?} \]
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Since both sides have \(\sqrt{3}\), we can compare the coefficients: \[ 2? + 6 = 14 \]
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Solve for \(?\): \[ 2? = 14 - 6 \] \[ 2? = 8 \] \[ ? = 4 \]
Now we can substitute back to confirm: \[ 4\sqrt{12} + 2\sqrt{27} = 4(2\sqrt{3}) + 6\sqrt{3} = 8\sqrt{3} + 6\sqrt{3} = 14\sqrt{3} \]
Thus, the complete expression is: \[ 4\sqrt{12} + 2\sqrt{27} = 14\sqrt{4} \]
Next, let's calculate \(8\sqrt{3}\): \[ 8\sqrt{3} \text{ remains as } 8\sqrt{3} \text{, and does not require further simplification.} \]
That completes the answer!
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