Given: \((5^{\sqrt{2}})^m=25\)

Apply the power of a power law to get: \((5^{m\sqrt{2}})=25\)

Rewrite \(25\) as \(5^2\) to get: \((5^{m\sqrt{2}})=5^2\)

So, it must be the case that: \(m\sqrt{2} = 2\)

Divide both sides of the equation by: \(\sqrt{2}\) to get: \(m = \frac{2}{\sqrt{2}}\)

This option does not appear among the answer choices, so we can create an equivalent fraction by multiplying numerator and denominator by \(\sqrt{2}\).

When we do so we get: \(m = \frac{2\sqrt{2}}{2}\)

Simplify to get: \(m = \sqrt{2}\)

Answer: C