Apply the Properties of Integer Exponents to generate equivalent expressions to 40 ⋅ 5−2(−2)−3

Question

with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)
40 ⋅ 5−2(−2)−3=

Answer 1

To solve the expression \( 40 \cdot 5^{-2} \cdot (-2)^{-3} \) using the properties of integer exponents, we start by rewriting the parts that have negative exponents.

1. Rewriting with Positive Exponents:
   • The term \( 5^{-2} \) can be rewritten as \( \frac{1}{5^2} \).
   • The term \( (-2)^{-3} \) can be rewritten as \( \frac{1}{(-2)^3} \).

Thus, we have:

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

Now substitute \( 5^2 = 25 \) and \( (-2)^3 = -8 \):

\[ 40 \cdot \frac{1}{25} \cdot \frac{1}{-8} \]

2. Combining the Expressions: Now we combine the fractions:

\[ 40 \cdot \frac{1}{25} \cdot \frac{1}{-8} = \frac{40}{25} \cdot \frac{1}{-8} \]

3. Calculating \( \frac{40}{25} \):

\[ \frac{40}{25} = \frac{8}{5} \]

Now we can plug this into the expression:

\[ \frac{8}{5} \cdot \frac{1}{-8} = \frac{8}{5} \cdot \frac{1}{-8} = \frac{8}{-40} = \frac{-1}{5} \]

4. Final Answer:

Thus, the simplified fraction with no remaining exponents is:

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