Apply the Properties of Integer Exponents to generate equivalent expressions to 4 raised to the 0 power ⋅ 5 raised to the −2 power/(−2) raised to the −3 power with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.

Answer 1

To simplify the expression \( \frac{4^0 \cdot 5^{-2}}{(-2)^{-3}} \) using the Properties of Integer Exponents, let's first deal with each part of the expression:

1. Evaluate \( 4^0 \): \[ 4^0 = 1 \]

2. Handle \( 5^{-2} \): \[ 5^{-2} = \frac{1}{5^2} = \frac{1}{25} \]

3. Handle \( (-2)^{-3} \): \[ (-2)^{-3} = \frac{1}{(-2)^3} = \frac{1}{-8} = -\frac{1}{8} \]

Now substituting back into the original expression: \[ \frac{4^0 \cdot 5^{-2}}{(-2)^{-3}} = \frac{1 \cdot \frac{1}{25}}{-\frac{1}{8}} \]

Let's simplify \( \frac{\frac{1}{25}}{-\frac{1}{8}} \): \[ \frac{\frac{1}{25}}{-\frac{1}{8}} = \frac{1}{25} \cdot -8 = -\frac{8}{25} \]

Thus, the expression simplifies to: \[ -\frac{8}{25} \]

This is the final result. Therefore, the answer is: \[ \boxed{-\frac{8}{25}} \]