Question

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

1 answer

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:

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

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