Question

Apply the properties of integer exponents to generate equivalent expressions that is four when the power of 0×5 with the power of -2 over and parentheses -2 when the power of -3 with only positive exponents then solve the expression your answer will be a simplified fraction with no remaining exponents

Answer 1

To create an equivalent expression from the given situation, let's break down the components step by step.

We start with what appears to be \(0.5^{-2}\) and \((-2)^{-3}\).

1. **Handling \(0.5^{-2}\)**:
- By the properties of exponents, \(a^{-n} = \frac{1}{a^n}\).
- Thus:
\[
0.5^{-2} = \frac{1}{0.5^2}
\]
- Calculating \(0.5^2\):
\[
0.5^2 = \frac{1}{4}
\]
- Therefore:
\[
0.5^{-2} = \frac{1}{\frac{1}{4}} = 4
\]

2. **Handling \((-2)^{-3}\)**:
- Again using the property of exponents:
\[
(-2)^{-3} = \frac{1}{(-2)^3}
\]
- Calculating \((-2)^3\):
\[
(-2)^3 = -8
\]
- Therefore:
\[
(-2)^{-3} = \frac{1}{-8}
\]

Now, substitute back:

We have the expression composed of:

\[
\frac{0.5^{-2}}{(-2)^{-3}} = \frac{4}{\frac{1}{-8}}
\]

3. **Calculating the final expression**:
- Dividing by a fraction is the same as multiplying by its reciprocal:
\[
\frac{4}{\frac{1}{-8}} = 4 \times -8 = -32
\]

Therefore, the final simplified answer is:

\[
\boxed{-32}
\]