To simplify the expression \(\sqrt{\frac{35}{343}}\), you can break it down as follows:
\[ \sqrt{\frac{35}{343}} = \frac{\sqrt{35}}{\sqrt{343}} \]
Next, simplify \(\sqrt{343}\):
\[ 343 = 7^3 \quad \text{so} \quad \sqrt{343} = \sqrt{7^3} = 7 \sqrt{7} \]
Now, substitute back into the expression:
\[ \sqrt{\frac{35}{343}} = \frac{\sqrt{35}}{7\sqrt{7}} \]
This fraction can further be simplified because \(35 = 5 \times 7\):
\[ \sqrt{35} = \sqrt{5 \times 7} = \sqrt{5} \sqrt{7} \]
So we can replace \(\sqrt{35}\) in the expression:
\[ \frac{\sqrt{5} \sqrt{7}}{7\sqrt{7}} = \frac{\sqrt{5}}{7} \]
Thus, the simplified expression for \(\sqrt{\frac{35}{343}}\) is:
\[ \frac{\sqrt{5}}{7} \]