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Question
These students were asked to simplify:  2⋅3−45−2

Adam wrote:  2⋅5234

Nick wrote:  522⋅34

Shane wrote:  2⋅3452

(1 point)
Responses

Nick is correct because the negative exponent in the numerator moves the whole numerator into the denominator and makes the exponent positive.
Nick is correct because the negative exponent in the numerator moves the whole numerator into the denominator and makes the exponent positive.

Adam is correct because the negative exponent moves the base to the other side of the fraction bar and makes the exponent positive.
Adam is correct because the negative exponent moves the base to the other side of the fraction bar and makes the exponent positive.

None of these are correct because the students did not add the exponents in the numerator and then subtract exponents since there is division.
None of these are correct because the students did not add the exponents in the numerator and then subtract exponents since there is divis…

1 answer

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.

  1. 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 \( \frac{1}{5^2} \).
  2. Rewriting the Expression: \[ 2 \cdot 3^{-4} \cdot 5^{-2} = 2 \cdot \frac{1}{3^4} \cdot \frac{1}{5^2} = \frac{2}{3^4 \cdot 5^2} \]

  3. 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.
  4. 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.