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

Question

Adam wrote: 2 ⋅ 5^2/3^4


Nick wrote: 5^2/2⋅3^4


Shane wrote: 2⋅3^4/5^2

A. None of these are correct because the students did not add the exponents in the numerator and then subtract exponents since there is division.
B. Shane is correct because all that needs to be done is to make the negative exponents positive.
C. Adam is correct because the negative exponent moves the base to the other side of the fraction bar and makes the exponent positive.
D. Nick is correct because the negative exponent in the numerator moves the whole numerator into the denominator and makes the exponent positive.

Answer 1

To simplify the expression \( 2 \cdot 3^{-4}/5^{-2} \), we first need to make all negative exponents positive by moving their bases to the opposite side of the fraction bar.

1. The term \( 3^{-4} \) in the numerator means we can move it to the denominator and it becomes \( 3^4 \).
2. The term \( 5^{-2} \) in the denominator means we can move it to the numerator and it becomes \( 5^2 \).

Thus, the expression transforms as follows:

\[ 2 \cdot 3^{-4} / 5^{-2} = 2 / 5^{-2} \cdot 3^{-4} = 2 \cdot 5^2 / 3^4 \]

Now let's examine each student's answer:

• Adam wrote: \( 2 \cdot 5^2 / 3^4 \) (This is correct.)
• Nick wrote: \( 5^2 / (2 \cdot 3^4) \) (This is incorrect.)
• Shane wrote: \( 2 \cdot 3^4 / 5^2 \) (This is also incorrect.)

Based on the analysis, Adam is correct.

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