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.
- Rewrite \( 5^{-2} \) as \( \frac{1}{5^2} \).
- 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:
\[ \boxed{-\frac{1}{5}} \]
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