c) \( \frac{(125 m)^{\frac{1}{3}}}{25\left(\mathrm{~m}^{-\frac{1}{3}}\right)^{2}} \)

To simplify the expression \( \frac{(125 m)^{\frac{1}{3}}}{25\left(\mathrm{~m}^{-\frac{1}{3}}\right)^{2}} \), let's break it down. The numerator \( (125 m)^{\frac{1}{3}} \) can be handled by taking the cube root of 125 and \( m \), resulting in \( 5m^{\frac{1}{3}} \). Now, focusing on the denominator, \( 25\left(\mathrm{~m}^{-\frac{1}{3}}\right)^{2} \) gives us \( 25m^{-\frac{2}{3}} \) after squaring the exponent. Now we rewrite the whole expression: \[ \frac{5m^{\frac{1}{3}}}{25m^{-\frac{2}{3}}} \] This can be simplified by cancelling the 5 and 25 to yield \( \frac{1}{5} \). Next, apply the properties of exponents for \( m^{\frac{1}{3}} \) divided by \( m^{-\frac{2}{3}} \): \[ m^{\frac{1}{3} - (-\frac{2}{3})} = m^{\frac{1}{3} + \frac{2}{3}} = m^{1} = m \] So the final result is: \[ \frac{1}{5}m \] Thus, the simplified expression is \( \frac{1}{5} m \).