Question

To simplify the expression \( 40 \cdot 5^{-2} \cdot (-2)^{-3} \) using the properties of integer exponents, we first need to rewrite the terms with negative exponents in terms of positive exponents.

1. Rewrite \( 5^{-2} \) as \( \frac{1}{5^2} \).
2. Rewrite \( (-2)^{-3} \) as \( \frac{1}{(-2)^3} \).

Now substituting these into the original expression:

\[
40 \cdot 5^{-2} \cdot (-2)^{-3} = 40 \cdot \frac{1}{5^2} \cdot \frac{1}{(-2)^3}
\]

Next, simplify this:

\[
= \frac{40}{5^2 \cdot (-2)^3}
\]

Now calculate \( 5^2 \) and \( (-2)^3 \):

- \( 5^2 = 25 \)
- \( (-2)^3 = -8 \)

Now substitute back:

\[
= \frac{40}{25 \cdot (-8)}
\]

Next, compute \( 25 \cdot (-8) \):

\[
= 25 \cdot (-8) = -200
\]

Therefore, the expression becomes:

\[
= \frac{40}{-200}
\]

To simplify \( \frac{40}{-200} \):

Dividing both the numerator and denominator by 40 gives:

\[
= \frac{1}{-5} = -\frac{1}{5}
\]

Thus, the final simplified result of the expression \( 40 \cdot 5^{-2} \cdot (-2)^{-3} \) is:

\[
\boxed{-\frac{1}{5}}
\]