\( \frac{\text { Simplify the following: }}{\frac{10^{x}+4 \cdot 2^{x}}{5^{2 x}+4 \cdot 5^{x}}} \)

To simplify \( \frac{10^{x}+4 \cdot 2^{x}}{5^{2 x}+4 \cdot 5^{x}} \), we can start by rewriting \( 10^{x} \) as \( (2 \cdot 5)^{x} = 2^{x} \cdot 5^{x} \). So, the numerator becomes: \[ 10^{x} + 4 \cdot 2^{x} = 2^{x} \cdot 5^{x} + 4 \cdot 2^{x} = 2^{x} (5^{x} + 4) \] Next, look at the denominator: \[ 5^{2x} + 4 \cdot 5^{x} = (5^{x})^{2} + 4 \cdot 5^{x} \] This can be factored as: \[ 5^{x}(5^{x} + 4) \] Now, substituting back into the original expression gives: \[ \frac{2^{x}(5^{x}+4)}{5^{x}(5^{x}+4)} \] Assuming \( 5^{x}+4 \) is not zero (which we can safely assume for values of \( x \)), we can cancel \( 5^{x}+4 \): \[ \frac{2^{x}}{5^{x}} = \left(\frac{2}{5}\right)^{x} \] Thus, the simplified expression is: \[ \left(\frac{2}{5}\right)^{x} \]