To solve the expression \( 2 \cdot 3^{-4} \cdot 5^{-2} \), we need to understand how negative exponents work and how to simplify the expression correctly.
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Understanding Negative Exponents:
- A negative exponent means that we take the reciprocal of the base and make the exponent positive. For example, \( a^{-n} = \frac{1}{a^n} \).
- Thus, \( 3^{-4} \) can be rewritten as \( \frac{1}{3^4} \), and \( 5^{-2} \) can be rewritten as \( \frac{1}{5^2} \).
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Rewriting the Expression: \[ 2 \cdot 3^{-4} \cdot 5^{-2} = 2 \cdot \frac{1}{3^4} \cdot \frac{1}{5^2} = \frac{2}{3^4 \cdot 5^2} \]
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Examining the Students' Responses:
- Adam wrote \( 2 \cdot 5^{2} \cdot 3^{4} \): This is incorrect because he did not correctly apply the negative exponents.
- Nick wrote \( 5^{2} \cdot 2 \cdot 3^{4} \): This is also incorrect due to similar reasons as Adam.
- Shane wrote \( 2 \cdot 3^{4} \cdot 5^{2} \): This is also incorrect for the above reasons.
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Conclusion of Correctness:
- All three students made errors in handling the negative exponents. The negative exponents should effectively move the terms to the denominator, so none of their answers are correct.
Therefore, the correct choice is: None of these are correct because the students did not add the exponents in the numerator and then subtract exponents since there is division.