Question

Question

These students were asked to simplify:  2⋅3^−4/5^−2

Adam wrote:  2⋅5^2/3^4

Nick wrote:  5^2/2⋅3^4

Shane wrote:  2⋅3^4/5^2

(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…

Answer 1

Let's evaluate the expression \( \frac{2 \cdot 3^{-4}}{5^{-2}} \) and analyze the students' responses.

1. Interpret the expression: \[ \frac{2 \cdot 3^{-4}}{5^{-2}} = 2 \cdot \frac{3^{-4}}{5^{-2}} = 2 \cdot 3^{-4} \cdot 5^2 \] Here, the negative exponent indicates that the base moves to the other side of the fraction and becomes positive.

2. Simplifying further: \[ = 2 \cdot \frac{5^2}{3^4} \] This can be organized to show: \[ = \frac{2 \cdot 5^2}{3^4} \]

3. Analyzing the students' answers:

   • Adam wrote: \( \frac{2 \cdot 5^2}{3^4} \) — this matches our derived expression.
   • Nick wrote: \( \frac{5^2}{2 \cdot 3^4} \) — this is incorrect because the 2 should be in the numerator with \( 5^2 \).
   • Shane wrote: \( \frac{2 \cdot 3^4}{5^2} \) — this is incorrect because it misplaces both the coefficients and the exponents.

4. Determining the correct response: Since Adam’s expression is equivalent to the correctly simplified form of the original expression, Adam is correct.

5. Conclusion: The appropriate response is:

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