2. \( \frac{5^{-3} \cdot 2^{4} \cdot 5^{0} \cdot 2}{2^{-1} \cdot 5^{-4}} \)

To simplify the expression \( \frac{5^{-3} \cdot 2^{4} \cdot 5^{0} \cdot 2}{2^{-1} \cdot 5^{-4}} \), we first simplify the numerator and denominator separately. **Numerator:** - \( 5^{-3} \cdot 5^{0} = 5^{-3} \cdot 1 = 5^{-3} \) - \( 2^{4} \cdot 2 = 2^{4} \cdot 2^{1} = 2^{5} \) - So, the numerator becomes \( 5^{-3} \cdot 2^{5} \). **Denominator:** - \( 2^{-1} \) remains as is, and \( 5^{-4} \) remains as is. - So, the denominator becomes \( 2^{-1} \cdot 5^{-4} \). Now, we can rewrite the entire expression as: \[ \frac{5^{-3} \cdot 2^{5}}{2^{-1} \cdot 5^{-4}} \] Next, we can split the fraction: \[ \frac{5^{-3}}{5^{-4}} \cdot \frac{2^{5}}{2^{-1}} \] Now simplify each part: - For the first part: \[ \frac{5^{-3}}{5^{-4}} = 5^{-3 - (-4)} = 5^{1} = 5 \] - For the second part: \[ \frac{2^{5}}{2^{-1}} = 2^{5 - (-1)} = 2^{5 + 1} = 2^{6} \] Putting it all together: \[ 5 \cdot 2^{6} \] Finally, to find the numerical value: \[ 2^{6} = 64 \\ \Rightarrow 5 \cdot 64 = 320 \] Thus, the simplified expression evaluates to: \[ \boxed{320} \]