Let's simplify the expression step-by-step using the properties of integer exponents.
The expression is:
\[ \frac{(7^{-3})^7 \cdot 7^{20}}{6^{-2}} \]
Step 1: Simplify \((7^{-3})^7\)
We use the power of a power property, which states that \((a^m)^n = a^{m \cdot n}\):
\[ (7^{-3})^7 = 7^{-3 \cdot 7} = 7^{-21} \]
Step 2: Combine \(7^{-21}\) and \(7^{20}\)
Now, we rewrite the expression as:
\[ \frac{7^{-21} \cdot 7^{20}}{6^{-2}} \]
Next, we can combine the terms in the numerator by using the product of powers property, which states that \(a^m \cdot a^n = a^{m + n}\):
\[ 7^{-21} \cdot 7^{20} = 7^{-21 + 20} = 7^{-1} \]
Step 3: Rewrite the expression
Now, we have:
\[ \frac{7^{-1}}{6^{-2}} \]
Step 4: Simplify \(6^{-2}\)
Next, by using the negative exponent property, \(a^{-m} = \frac{1}{a^m}\):
\[ 6^{-2} = \frac{1}{6^2} = \frac{1}{36} \]
Step 5: Rewrite the fraction
We can rewrite the entire expression as follows:
\[ \frac{7^{-1}}{6^{-2}} = \frac{7^{-1}}{\frac{1}{36}} = 7^{-1} \cdot 36 \]
Step 6: Apply the negative exponent property again
Now, we convert \(7^{-1}\):
\[ 7^{-1} = \frac{1}{7} \]
Thus, we have:
\[ \frac{1}{7} \cdot 36 = \frac{36}{7} \]
Final Answer
Hence, the final answer is:
\[ \boxed{\frac{36}{7}} \]
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