To simplify \( 3\sqrt{686} \), we first need to simplify the square root of 686.
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Factor 686 into its prime factors:
- \( 686 \) is even, so divide by \( 2 \): \[ 686 \div 2 = 343 \]
- Next, factor \( 343 \): \[ 343 = 7 \times 49 = 7 \times 7 \times 7 = 7^3 \]
- So, \( 686 = 2 \times 7^3 \).
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Use the property of square roots: \[ \sqrt{686} = \sqrt{2 \times 7^3} = \sqrt{2} \times \sqrt{7^3} = \sqrt{2} \times 7 \times \sqrt{7} = 7\sqrt{14} \] (since \( \sqrt{7^3} = 7\sqrt{7} \) and \( \sqrt{2} \times 7 \times \sqrt{7} = 7\sqrt{14} \))
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Substitute back into the original expression: \[ 3\sqrt{686} = 3(7\sqrt{14}) = 21\sqrt{14} \]
Thus, the simplified form of \( 3\sqrt{686} \) is: \[ \boxed{21\sqrt{14}} \]