Question

Apply the Properties of Integer Exponents to generate equivalent expressions to 4 to the zero power 0 ⋅ 5 2 to the second power−2=5-2/(−2) to the third power−3
with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.(1 point)

Answer 1

wrong

Answer 2

To solve the given expression, we need to apply properties of integer exponents and convert everything into a form that contains only positive exponents.

The expression is:

\[
4^0 \cdot 5^2 \cdot 2^{-2} = \frac{5^{-2}}{(-2)^{-3}}
\]

We can start simplifying each part:

1. **Calculate \(4^0\)**:
\[
4^0 = 1
\]

2. **Simplify \(5^2\)**:
\[
5^2 = 25
\]

3. **Rewrite \(2^{-2}\)** using positive exponents:
\[
2^{-2} = \frac{1}{2^2} = \frac{1}{4}
\]

Now the left side becomes:
\[
1 \cdot 25 \cdot \frac{1}{4} = \frac{25}{4}
\]

Now let's simplify the right side, which is \(\frac{5^{-2}}{(-2)^{-3}}\):

1. **Rewrite \(5^{-2}\)** using positive exponents:
\[
5^{-2} = \frac{1}{5^2} = \frac{1}{25}
\]

2. **Rewrite \((-2)^{-3}\)** using positive exponents:
\[
(-2)^{-3} = \frac{1}{(-2)^3} = \frac{1}{-8}
\]

Now the right side becomes:
\[
\frac{5^{-2}}{(-2)^{-3}} = \frac{\frac{1}{25}}{\frac{1}{-8}} = \frac{1}{25} \cdot \left(-8\right) = -\frac{8}{25}
\]

Now we equate both sides:
\[
\frac{25}{4} = -\frac{8}{25}
\]

These fractions do not equal each other, which indicates we should have set up the expressions carefully according to the given equality.

We may want to recalculate from the left without making any errors:

The expression \(4^0 \cdot 5^2 \cdot 2^{-2}\) simplifies to:
- \(4^0 = 1\)
- \(5^2 = 25\)
- \(2^{-2} = \frac{1}{4}\)

Thus:
\[
1 \cdot 25 \cdot \frac{1}{4} = \frac{25}{4}
\]

The complete left side evaluates to \(\frac{25}{4}\) accurately.

This value doesn't thus equal to the right side if set up from the main equality. Therefore, the final simplified result based on the left-hand side of the original expression results in:

\[
\frac{25}{4}
\]

Thus your final simplified fraction is:
\[
\boxed{\frac{25}{4}}
\]

Therefore, the computation indicates that simplifying contextually connects to positive exponents and combined evaluations yielding consistent results when isolating components carefully.

Answer 3

Let's carefully analyze the expression step-by-step and make sure we apply the properties of integer exponents correctly, generating equivalent expressions with only positive exponents, and solving the expression properly.

Given:
\[
4^0 \cdot 5^2 \cdot 2^{-2} = \frac{5^{-2}}{(-2)^{-3}}
\]

### Left Side Simplification

1. **Evaluate \(4^0\)**:
\[
4^0 = 1
\]

2. **Calculate \(5^2\)**:
\[
5^2 = 25
\]

3. **Convert \(2^{-2}\) to positive exponents**:
\[
2^{-2} = \frac{1}{2^2} = \frac{1}{4}
\]

Now combining these, we have:
\[
1 \cdot 25 \cdot \frac{1}{4} = \frac{25}{4}
\]

### Right Side Simplification

Now let's simplify the right-hand side:
\[
\frac{5^{-2}}{(-2)^{-3}}
\]

4. **Convert \(5^{-2}\)**:
\[
5^{-2} = \frac{1}{5^2} = \frac{1}{25}
\]

5. **Convert \((-2)^{-3}\)**:
\[
(-2)^{-3} = \frac{1}{(-2)^3} = \frac{1}{-8}
\]

Now substituting into the expression gives us:
\[
\frac{5^{-2}}{(-2)^{-3}} = \frac{\frac{1}{25}}{\frac{1}{-8}}
\]
This can be rewritten as:
\[
\frac{1}{25} \cdot (-8) = -\frac{8}{25}
\]

### Final Equation

Now we have two sides to compare:
Left Side:
\[
\frac{25}{4}
\]

Right Side:
\[
-\frac{8}{25}
\]

### Conclusion

The two expressions \(\frac{25}{4}\) and \(-\frac{8}{25}\) indeed are not equal.

### Resolving the initial value

To resolve the original division and multiplication contextually, we simplify where applicable. Notably, the left-hand value is maintained as \(\frac{25}{4}\) while the right needs careful evaluative checks.

Thus, the overall simplified result based on the left expression remains:

\[
\boxed{\frac{25}{4}}
\]

If there are contextual mismatches in the reference expression provided versus expectations, please clarify so an adjustment can be made. Thank you for your patience!