\[\sqrt{38416} = \sqrt{2 \cdot 19208} =\]
\[= \sqrt{2 \cdot 2 \cdot 9604} =\]
\[= \sqrt{2 \cdot 2 \cdot 2 \cdot 4802} =\]
\[= \sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 2401} =\]
\[= \sqrt{2^{4} \cdot 7 \cdot 343} =\]
\[= \sqrt{2^{4} \cdot 7^{2} \cdot 49} = 2^{2} \cdot 7^{2} = 196\]