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

To determine which student's simplification is correct, let's start with the expression given:

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

1. Convert negative exponents:
   • The term \(3^{-4}\) can be rewritten as \(\frac{1}{3^4}\).
   • The term \(5^{-2}\) can be rewritten as \(5^2\) in the numerator.

So, we can rewrite the expression as:

\[ \frac{2 \cdot \frac{1}{3^4}}{5^{-2}} = \frac{2}{3^4} \cdot 5^2 = \frac{2 \cdot 5^2}{3^4} \]

2. Rewrite in standard form: This gives us:

\[ \frac{2 \cdot 5^2}{3^4} \]

Now, let's evaluate each student's answer:

• 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.

3. Statement evaluations:
   • Adam's statement is correct in that he correctly moved the negative exponents and simplified.
   • Shane's statement is incorrect because it misrepresents the relationship of the terms.
   • Nick's statement is incorrect in both simplification and reasoning.

Thus, the only correct answer is Adam's:

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