$\sqrt{3 w+14}=\sqrt{7 w-14}$

Step 1 ：Given the equation \(\sqrt{3 w+14}=\sqrt{7 w-14}\)

Step 2 ：Square both sides of the equation to eliminate the square roots, resulting in \(3w + 14 = 7w - 14\)

Step 3 ：Rearrange the equation to isolate w, resulting in \(4w = 28\)

Step 4 ：Divide both sides by 4 to solve for w, resulting in \(w = 7\)

Step 5 ：Final Answer: \(\boxed{7}\)