Question Apply the Properties of Integer Exponents to generate equivalent expressions to (7^−3)^7 ⋅ 7^20/6^−2 with only positive exponents, then solve the expression. Provide the answer as an improper fraction

To simplify the expression \((7^{-3})^7 \cdot \frac{7^{20}}{6^{-2}}\) and generate equivalent expressions with only positive exponents, we can use the properties of integer exponents.

1. **Apply the power of a power property**:

\[
(7^{-3})^7 = 7^{-3 \cdot 7} = 7^{-21}
\]

2. **Next, we rewrite the expression**:

\[
7^{-21} \cdot \frac{7^{20}}{6^{-2}}
\]

3. **Rewrite the denominator with positive exponents**:

\(6^{-2} = \frac{1}{6^2}\), therefore:

\[
\frac{7^{20}}{6^{-2}} = 7^{20} \cdot 6^2
\]

This allows us to rewrite the whole expression:

\[
7^{-21} \cdot 7^{20} \cdot 6^2
\]

4. **Combine the powers of \(7\)**:

\[
7^{-21 + 20} \cdot 6^2 = 7^{-1} \cdot 6^2
\]

5. **Changing \(7^{-1}\) to positive exponent**:

\[
7^{-1} = \frac{1}{7}
\]

Therefore, the expression is:

\[
\frac{1}{7} \cdot 6^2
\]

6. **Calculate \(6^2\)**:

\[
6^2 = 36
\]

So, we have:

\[
\frac{1}{7} \cdot 36 = \frac{36}{7}
\]

Thus, the final answer is:

\[
\frac{36}{7}
\]

This is the expression simplified with only positive exponents, and the answer in improper fraction form is:

\[
\boxed{\frac{36}{7}}
\]

Bot, can you be more clear?

Sure! Let's break it down step-by-step to clarify the process of simplifying the expression \((7^{-3})^7 \cdot \frac{7^{20}}{6^{-2}}\) and ensuring that we only use positive exponents.

### Step 1: Simplify \((7^{-3})^7\)
Using the power of a power property:
\[
(7^{-3})^7 = 7^{-3 \cdot 7} = 7^{-21}
\]

### Step 2: Rewrite the Expression
Now we plug this back into our expression:
\[
7^{-21} \cdot \frac{7^{20}}{6^{-2}}
\]

### Step 3: Handle the Denominator
Recall that \(6^{-2}\) can be rewritten with a positive exponent:
\[
6^{-2} = \frac{1}{6^2}
\]
This means that dividing by \(6^{-2}\) is the same as multiplying by \(6^2\):
\[
\frac{7^{20}}{6^{-2}} = 7^{20} \cdot 6^2
\]

### Step 4: Combine Everything
Now substitute this back into our expression:
\[
7^{-21} \cdot 7^{20} \cdot 6^2
\]

### Step 5: Combine Powers of \(7\)
Now we can combine the powers of \(7\):
\[
7^{-21 + 20} = 7^{-1}
\]
So the expression becomes:
\[
7^{-1} \cdot 6^2
\]

### Step 6: Convert to Positive Exponents
Convert \(7^{-1}\) to a positive exponent:
\[
7^{-1} = \frac{1}{7}
\]

### Step 7: Calculate \(6^2\)
Now calculate \(6^2\):
\[
6^2 = 36
\]

### Step 8: Combine All Parts
Putting it all together:
\[
\frac{1}{7} \cdot 36 = \frac{36}{7}
\]

### Final Answer
The final expression is:
\[
\boxed{\frac{36}{7}}
\]

This is the result expressed as an improper fraction with all positive exponents. Let me know if you need further clarification!