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: $$\boxed{{\displaystyle -\frac{8}{25}}}$$