To simplify the expression \(59 \cdot 5^{-13}\) and make it have only positive exponents, let's apply properties of integer exponents.
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Start with the original expression: \[ 59 \cdot 5^{-13} \]
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We can rewrite the expression using the fact that \(5^{-13} = \frac{1}{5^{13}}\): \[ 59 \cdot 5^{-13} = 59 \cdot \frac{1}{5^{13}} = \frac{59}{5^{13}} \]
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Now consider the options provided:
- 54
- \(5^4\) (it's unclear what the "baseline" indicates, but this is just \(5^4\))
- \( \frac{5^9}{5^{-13}} \)
- \( 59 \cdot \frac{1}{5^4} \)
- \( 15 \cdot 22\)
Next, let's check each response to see if it matches our simplified expression:
- 54: This does not match.
- \(5^4\): This does not match.
- \( \frac{5^9}{5^{-13}} \): This can be simplified using the property of exponents: \[ \frac{5^9}{5^{-13}} = 5^{9 - (-13)} = 5^{9 + 13} = 5^{22} \] This does not match.
- \( 59 \cdot \frac{1}{5^4} \): This does not match since it has \(5^4\) instead of \(5^{13}\).
- \( 15 \cdot 22\): This is also irrelevant.
None of the provided options represent \( \frac{59}{5^{13}} \), but option \( \frac{5^9}{5^{-13}} \), which simplifies to \(5^{22}\), is presented in a way that is a mathematical transformation.
However, it seems there is an error in the options provided or in my interpretation, as none of the responses accurately match our simplified expression. The most candidate-like option would be one particularly indicating a transformation involving \(5^{-13}\) if we consider interaction with bases.
Therefore, if the expression must strictly represent \( \frac{59}{5^{13}}\), we have come to the conclusion that it is best represented simply as this ratio format.
If any specific representation for clarity is needed, do let me know. Out of internal checking, our best alignments with transformations are limited from the provided choices.
3 answers