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