Question

Apply the Properties of Integer Exponents to generate equivalent expressions to (4^3 times 7^-2)^-5/7^9 times 4^-13 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.

Answer 1

thats wrong

Answer 2

To simplify the expression \((4^3 \cdot 7^{-2})^{-5} \cdot \frac{1}{7^9} \cdot 4^{-13}\) and rewrite it with only positive exponents, we will apply the properties of integer exponents step by step.

1. **Distributing the exponent on \((4^3 \cdot 7^{-2})^{-5}\)**:
\[
(4^3 \cdot 7^{-2})^{-5} = 4^{3 \cdot -5} \cdot 7^{-2 \cdot -5} = 4^{-15} \cdot 7^{10}.
\]

2. **Rewrite the entire expression**:
\[
4^{-15} \cdot 7^{10} \cdot \frac{1}{7^9} \cdot 4^{-13}.
\]

3. **Convert \(\frac{1}{7^9}\) to a negative exponent**:
\[
4^{-15} \cdot 7^{10} \cdot 7^{-9} \cdot 4^{-13}.
\]

4. **Combining the \(4\) terms**:
\[
4^{-15} \cdot 4^{-13} = 4^{-15 + (-13)} = 4^{-28}.
\]

5. **Combining the \(7\) terms**:
\[
7^{10} \cdot 7^{-9} = 7^{10 + (-9)} = 7^{1}.
\]

6. **Putting it all together**:
\[
4^{-28} \cdot 7^{1} = \frac{7}{4^{28}}.
\]

7. **Rewrite the expression with positive exponents**:
\[
\frac{7}{4^{28}}.
\]

The final answer is:
\[
\frac{7}{4^{28}}.
\]

Answer 3

Let's go through the steps carefully to simplify the expression \((4^3 \cdot 7^{-2})^{-5} \cdot \frac{1}{7^9} \cdot 4^{-13}\) correctly and ensure we have only positive exponents at the end.

### Step 1: Simplify \((4^3 \cdot 7^{-2})^{-5}\)

Using the properties of exponents:
\[
(4^3 \cdot 7^{-2})^{-5} = (4^3)^{-5} \cdot (7^{-2})^{-5} = 4^{-15} \cdot 7^{10}
\]

### Step 2: Combine with the rest of the expression

Now replace \(\frac{1}{7^9}\) with \(7^{-9}\):
\[
4^{-15} \cdot 7^{10} \cdot 7^{-9} \cdot 4^{-13}
\]

### Step 3: Combine like terms for the bases

**Combining the \(4\) terms:**
\[
4^{-15} \cdot 4^{-13} = 4^{-15 - 13} = 4^{-28}
\]

**Combining the \(7\) terms:**
\[
7^{10} \cdot 7^{-9} = 7^{10 - 9} = 7^{1}
\]

### Step 4: Putting it all together

The combined expression becomes:
\[
4^{-28} \cdot 7^{1} = 7 \cdot 4^{-28}
\]

### Step 5: Rewrite with positive exponents

To write \(4^{-28}\) with a positive exponent:
\[
7 \cdot \frac{1}{4^{28}} = \frac{7}{4^{28}}
\]

### Final Answer

The final simplified expression is:
\[
\frac{7}{4^{28}}
\]

If you have a specific part that seems incorrect or you're looking for something different, please let me know!