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

Question

Adam wrote: 2 ⋅ 5^2 over 3^4


Nick wrote: 5^2 over 2 ⋅ 3^4

Shane wrote: 2 ⋅ 3^4 over 5^2

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

Shane is correct because all that needs to be done is to make the negative exponents positive.

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

Answer 1

Let’s analyze the expression \( \frac{2 \cdot 3^{-4}}{5^{-2}} \) and simplify it step by step.

1. Start with the original expression: \[ \frac{2 \cdot 3^{-4}}{5^{-2}} \]

2. Apply the rule that states a negative exponent means taking the reciprocal:

   • \( 3^{-4} \) becomes \( \frac{1}{3^4} \)
   • \( 5^{-2} \) becomes \( \frac{1}{5^2} \)

   Rewriting the expression gives us: \[ 2 \cdot 3^{-4} \cdot 5^2 = 2 \cdot 5^2 \cdot \frac{1}{3^4} = \frac{2 \cdot 5^2}{3^4} \]

Thus, the correct simplified version of the expression is: \[ \frac{2 \cdot 5^2}{3^4} \]

Now let's evaluate the responses from the students:

• Adam: Wrote \( \frac{2 \cdot 5^2}{3^4} \) – This is correct.
• Nick: Wrote \( \frac{5^2}{2 \cdot 3^4} \) – This is incorrect.
• Shane: Wrote \( \frac{2 \cdot 3^4}{5^2} \) – This is incorrect.

Now, regarding the justifications:

• Adam is correct because he correctly moved the negative exponent bases, yielding \( \frac{2 \cdot 5^2}{3^4} \).
• Shane is incorrect because he miscalculated the positions of the bases while simplifying.
• Nick is incorrect as well; the numerators and denominators were not correctly manipulated.
• The statement None of these are correct is false because Adam's answer is indeed correct.

So, the final conclusion is: Adam is correct.