To simplify the expression \( (7^{-3})^7 \cdot 7^{206 - 2} \) using the properties of integer exponents, we will apply the laws of exponents step by step.
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Applying the power of a power property: \[ (a^m)^n = a^{m \cdot n} \] Thus, we have: \[ (7^{-3})^7 = 7^{-3 \cdot 7} = 7^{-21} \]
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Simplifying the second part: \[ 7^{206 - 2} = 7^{204} \]
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Combining the expressions: Now we can combine the two parts: \[ 7^{-21} \cdot 7^{204} \]
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Applying the product of powers property: \[ a^m \cdot a^n = a^{m+n} \] Therefore: \[ 7^{-21 + 204} = 7^{183} \]
Now we have simplified our expression to \( 7^{183} \).
To express this in terms of an improper fraction: \[ 7^{183} = \frac{7^{183}}{1} \]
So the final answer, expressed as an improper fraction, is: \[ \boxed{\frac{7^{183}}{1}} \]