Question

Simplying radicals
3 Square root of 686

Answer 1

To simplify \( 3\sqrt{686} \), we first need to simplify the square root of 686.

1. **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, \( 686 = 2 \times 7^3 \).

2. **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 \( \sqrt{2} \times 7 \times \sqrt{7} = 7\sqrt{14} \))

3. **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:
\[
\boxed{21\sqrt{14}}
\]