To simplify the expression \( 40 \cdot 5^{-2} \cdot (-2)^{-3} \) using the properties of exponents, let's first rewrite the negative exponents as positive exponents.
- Rewrite \( 5^{-2} \) as \( \frac{1}{5^2} \).
- Rewrite \( (-2)^{-3} \) as \( \frac{1}{(-2)^3} \).
Now we can rewrite the expression:
\[ 40 \cdot 5^{-2} \cdot (-2)^{-3} = 40 \cdot \frac{1}{5^2} \cdot \frac{1}{(-2)^3} \]
- Calculate \( 5^2 \) and \( (-2)^3 \):
\[ 5^2 = 25 \] \[ (-2)^3 = -8 \]
- Substitute these values back into the expression:
\[ 40 \cdot \frac{1}{25} \cdot \frac{1}{-8} = \frac{40}{25 \cdot -8} \]
- Calculate \( 25 \cdot -8 \):
\[ 25 \cdot -8 = -200 \]
- So now our expression looks like:
\[ \frac{40}{-200} \]
- Simplify the fraction:
\[ \frac{40 \div 40}{-200 \div 40} = \frac{1}{-5} = -\frac{1}{5} \]
Thus, the simplified fraction of the expression \( 40 \cdot 5^{-2} \cdot (-2)^{-3} \) is
\[ -\frac{1}{5} \]