To solve the expression \((7^{-3})^7 \cdot 7^{206 - 2}\), we will first apply the properties of integer exponents to simplify it step by step.
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Simplify \((7^{-3})^7\): Using the property \((a^m)^n = a^{m \cdot n}\), we can rewrite this section:
\[ (7^{-3})^7 = 7^{-3 \cdot 7} = 7^{-21} \]
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Simplify \(7^{206 - 2}\): First, calculate \(206 - 2\):
\[ 206 - 2 = 204 \]
Thus,
\[ 7^{206 - 2} = 7^{204} \]
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Combine the two parts: Now we combine \(7^{-21}\) and \(7^{204}\):
\[ 7^{-21} \cdot 7^{204} \]
Using the property \(a^m \cdot a^n = a^{m+n}\), we have:
\[ 7^{-21 + 204} = 7^{183} \]
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Write in terms of positive exponent: To have a positive exponent, we express \(7^{183}\) as:
\[ 7^{183} = \frac{1}{7^{-183}} = \frac{1}{7^{-183}} = \frac{1}{7^{183}} \]
Now we have the simplified expression as \(7^{183}\).
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Final result: The expression \(7^{183}\) is already in a clear form for the answer, but to express it as a fraction:
\[ \text{Final Answer:} \quad \frac{1}{7^{183}} \]
This result is in improper fraction form with no exponents or decimals, and it meets the requirement as specified in the problem statement.
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