Question

Apply the Properties of Integer Exponents to generate equivalent expressions to
4^0 x 5^-2/ (-2)^-3
with positive exponents only, then solve the expression

Answer 1

CHAT GPT. 0, dummy, you got it wrong

Answer 2

To simplify the expression \( \frac{4^0 \times 5^{-2}}{(-2)^{-3}} \) using the properties of integer exponents, we first rewrite each term with positive exponents.

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

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

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

Now, substitute these values back into the expression:

\[
\frac{4^0 \times 5^{-2}}{(-2)^{-3}} = \frac{1 \times \frac{1}{25}}{\frac{1}{-8}} = \frac{\frac{1}{25}}{\frac{1}{-8}}
\]

4. **Dividing fractions**: To divide by a fraction, multiply by its reciprocal:

\[
\frac{\frac{1}{25}}{\frac{1}{-8}} = \frac{1}{25} \times (-8) = -\frac{8}{25}
\]

Thus, the expression \( \frac{4^0 \times 5^{-2}}{(-2)^{-3}} \), simplified to have only positive exponents, results in:

\[
-\frac{8}{25}
\]

Final result:
\[
\boxed{-\frac{8}{25}}
\]

Answer 3

Let’s go through the calculation again step by step to ensure accuracy.

We are given the expression:

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

### Step 1: Simplify \(4^0\)

\[
4^0 = 1
\]

### Step 2: Simplify \(5^{-2}\)

Using the property of negative exponents:

\[
5^{-2} = \frac{1}{5^2} = \frac{1}{25}
\]

### Step 3: Simplify \((-2)^{-3}\)

Using the property of negative exponents:

\[
(-2)^{-3} = \frac{1}{(-2)^3}
\]
Calculating \((-2)^3\):

\[
(-2)^3 = -8 \quad \text{so} \quad (-2)^{-3} = \frac{1}{-8}
\]

### Step 4: Substitute these values into the expression

Now substituting back into the original expression:

\[
\frac{4^0 \times 5^{-2}}{(-2)^{-3}} = \frac{1 \times \frac{1}{25}}{\frac{1}{-8}} = \frac{\frac{1}{25}}{\frac{1}{-8}}
\]

### Step 5: Dividing fractions

Dividing by a fraction means multiplying by its reciprocal:

\[
\frac{\frac{1}{25}}{\frac{1}{-8}} = \frac{1}{25} \times \left(-8\right) = -\frac{8}{25}
\]

### Final Result

The simplified expression, with all positive exponents, is:

\[
-\frac{8}{25}
\]

If any specific part of the process seems incorrect, please let me know! Otherwise, this confirms the evaluation of the expression as \( \boxed{-\frac{8}{25}} \).