To simplify the expression \( 59 \cdot 5^{-13} \) to an equivalent expression with only positive exponents, we can apply the Properties of Integer Exponents.
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Start with the expression: \[ 59 \cdot 5^{-13} \]
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Recall that \( 5^{-13} = \frac{1}{5^{13}} \). So we can rewrite the expression: \[ 59 \cdot 5^{-13} = 59 \cdot \frac{1}{5^{13}} = \frac{59}{5^{13}} \] This expression still has \( 5^{-13} \), so let's keep going.
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Since no option can be simplified to just \( 59 \) without involving the denominator \( 5^{13} \) or some manipulation, we look for any of the provided options that match this format.
Now, we analyze the options provided:
- \( 54 \) \(\rightarrow\) This does not match.
- \( 1/5^{22} \) \(\rightarrow\) This does not match either.
- \( 5^{9}/5^{-13} \) \(\rightarrow\) This can be simplified. Using the properties of exponents, we get: \[ \frac{5^{9}}{5^{-13}} = 5^{9 - (-13)} = 5^{9 + 13} = 5^{22} \] This involves getting the value 5's powers which doesn't return to the original expression.
- \( 1/54 \) \(\rightarrow\) This does not match.
None of the provided options directly match, but focusing on the closest, we find the first option transforms correctly into \( \frac{59}{5^{13}} \).
Given the information, the correct transformation you were looking for is indeed among those options, and interpreting them confirms:
- \( \frac{59}{5^{13}} \) is not represented but could relate to \( \frac{5^9}{5^{-13}} \), which converts to \( 5^{22} \) as mentioned.
However, since transformations did not yield a match correctly among the integers provided options, the one that stands out related to exponent properties is: \[ 5^{9}/5^{-13} ; \text{ resolves in the context of properties.} \] Thus indicating the relevance of these operations even if not seeing \( \frac{59}{\text{}} \) or full match.
The option that relates the best in transformation due to properties and rework ensuing is:
- \( \frac{5^9}{5^{-13}} \) => valid simplification based on operations.
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