c) \( \frac{125 \cdot\left(5^{3}\right)^{2}}{\left(5^{2}\right)^{4}}= \)

To solve the expression \( \frac{125 \cdot\left(5^{3}\right)^{2}}{\left(5^{2}\right)^{4}} \), we start by simplifying each part. First, recognize that \( 125 \) can be expressed as \( 5^{3} \). Now rewriting the expression gives us: \[ \frac{5^{3} \cdot (5^3)^2}{(5^2)^4} \] Next, we apply the power of a power rule \( (a^m)^n = a^{m \cdot n} \): \[ = \frac{5^{3} \cdot 5^{6}}{5^{8}} \] Now, combine the exponents in the numerator: \[ = \frac{5^{3 + 6}}{5^{8}} = \frac{5^{9}}{5^{8}} \] Using the quotient of powers rule \( \frac{a^m}{a^n} = a^{m-n} \): \[ = 5^{9 - 8} = 5^{1} = 5 \] Thus, the final answer is \( 5 \).