Apply the Properties of Integer Exponents to generate equivalent expressions to 40 ⋅ 5−2(−2)−3with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)

Answer 1

To simplify the expression \( 40 \cdot 5^{-2} \cdot (-2)^{-3} \) using the properties of integer exponents:

1. Convert negative exponents to positive:
   • For \( 5^{-2} \), we can write it as \( \frac{1}{5^2} \).
   • For \( (-2)^{-3} \), we can write it as \( \frac{1}{(-2)^3} \).

So, we rewrite the expression: \[ 40 \cdot \frac{1}{5^2} \cdot \frac{1}{(-2)^3} \]

2. Calculate the powers:
   • \( 5^2 = 25 \)
   • \( (-2)^3 = -8 \)

Now we have: \[ 40 \cdot \frac{1}{25} \cdot \frac{1}{-8} \]

3. Combine the fractions: \[ = 40 \cdot \frac{1}{25 \cdot (-8)} = 40 \cdot \frac{1}{-200} = \frac{40}{-200} \]

4. Simplify the fraction: \(\frac{40}{-200} = \frac{40 \div 40}{-200 \div 40} = \frac{1}{-5} = -\frac{1}{5}\)

Thus, the simplified expression is: \[ \boxed{-\frac{1}{5}} \]