To simplify the expression \((7^{-3}) \cdot 7^{20} / 6^{-2}\) while applying the properties of integer exponents and ensuring all exponents are positive, follow these steps:
Step 1: Simplify the powers of 7
Using the product of powers property, which states that \(a^m \cdot a^n = a^{m+n}\), we can combine the two powers of 7:
\[ 7^{-3} \cdot 7^{20} = 7^{-3 + 20} = 7^{17} \]
Step 2: Simplify the expression with respect to \(6^{-2}\)
The expression now looks like this:
\[ \frac{7^{17}}{6^{-2}} \]
Using the property that \(a^{-n} = \frac{1}{a^n}\), we can rewrite \(6^{-2}\):
\[ 6^{-2} = \frac{1}{6^2} \]
Thus, the expression can be rewritten as:
\[ \frac{7^{17}}{\frac{1}{6^2}} = 7^{17} \cdot 6^2 \]
Step 3: Rewrite the expression with positive exponents
Now we have:
\[ 7^{17} \cdot 6^2 = 7^{17} \cdot 36 \]
Step 4: Write the final expression
Now combine them together:
\[ 7^{17} \cdot 36 \]
Step 5: Final expression as an improper fraction
The final answer can be expressed as:
\[ \frac{36 \cdot 7^{17}}{1} \]
So the simplified expression, as an improper fraction, is:
\[ \frac{36 \cdot 7^{17}}{1} \]
This is the final answer, and it can also be presented as:
\[ \frac{36 \cdot 7^{17}}{1} \rightarrow 36 \cdot 7^{17} \]
However, in the strict sense of the question's request for an improper fraction format, it is:
\[ \frac{36 \cdot 7^{17}}{1} \] as the answer.