To determine who correctly used the Power Rule of Exponents to create an equivalent expression for \((153)^9\), let's evaluate the given expressions against the original expression.
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Power Rule of Exponents: The Power Rule states that \((a^m)^n = a^{m \cdot n}\). So when we have an expression such as \((153)^9\), it stays as it is, but we need to check if any conversions to numerical values match correctly.
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Evaluating Amy's Expression:
- Amy wrote \(1527\).
- \(153^9\) is significantly larger than \(1527\).
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Evaluating Lin's Expression:
- Lin wrote \(153\).
- This does not represent \((153)^9\), but rather just \(153\) itself, so it is not equivalent.
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Evaluating Miguel's Expression:
- Miguel wrote \(1512\).
- Similar to Amy's case, \(153^9\) is much larger than \(1512\).
Based on this analysis:
- Amy's and Miguel's expressions are all incorrect in that they do not equate to \((153)^9\).
- Lin's expression \(153\) also does not represent an equivalent expression for \((153)^9\), but is just a numerical representation of the base.
Therefore, none of them correctly expressed \((153)^9\) using the Power Rule of Exponents. All of them made mistakes, and the correct equivalent expression should have remained as \((153)^9\) or evaluated correctly without losing the exponent.
The answer is that none of them correctly used the Power Rule of Exponents.
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