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.
Shane is correct because all that needs to be done is to make the negative exponents positive.
Shane is correct because all that needs to be done is to make the negative exponents 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 division.
Skip to navigation
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.
Shane is correct because all that needs to be done is to make the negative exponents positive.
Shane is correct because all that needs to be done is to make the negative exponents 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 division.
Skip to navigation
Answers
GPT-4o mini
To simplify the expression \( \frac{2 \cdot 3^{-4}}{5^{-2}} \), let's address the negative exponents first.
1. **Understanding Negative Exponents**:
- \( a^{-n} = \frac{1}{a^n} \)
- To eliminate negative exponents, we can move them to the other side of the fraction bar, which changes their sign.
Using the properties of negative exponents:
- \( 3^{-4} = \frac{1}{3^4} \) (which means we can move it to the denominator)
- \( 5^{-2} = \frac{1}{5^2} \) (which means we can move it to the numerator)
Now, rewriting the expression:
\[
\frac{2 \cdot 3^{-4}}{5^{-2}} = \frac{2}{5^{-2}} \cdot \frac{1}{3^4} = 2 \cdot 5^2 \cdot \frac{1}{3^4} = \frac{2 \cdot 5^2}{3^4}
\]
2. **Comparing with Students' Answers**:
- **Adam's answer**: \( \frac{2 \cdot 5^2}{3^4} \) - this is correct.
- **Nick's answer**: \( \frac{5^2}{2 \cdot 3^4} \) - this is incorrect; he moved the \( 2 \) to the denominator.
- **Shane's answer**: \( \frac{2 \cdot 3^4}{5^2} \) - this is incorrect; he moved \( 3^{-4} \) in the wrong direction and didn't simplify correctly.
3. **Correct Answer**:
The valid answer is Adam's. Therefore, the correct response would be:
**"Adam is correct because the negative exponent moves the base to the other side of the fraction bar and makes the exponent positive."**
1. **Understanding Negative Exponents**:
- \( a^{-n} = \frac{1}{a^n} \)
- To eliminate negative exponents, we can move them to the other side of the fraction bar, which changes their sign.
Using the properties of negative exponents:
- \( 3^{-4} = \frac{1}{3^4} \) (which means we can move it to the denominator)
- \( 5^{-2} = \frac{1}{5^2} \) (which means we can move it to the numerator)
Now, rewriting the expression:
\[
\frac{2 \cdot 3^{-4}}{5^{-2}} = \frac{2}{5^{-2}} \cdot \frac{1}{3^4} = 2 \cdot 5^2 \cdot \frac{1}{3^4} = \frac{2 \cdot 5^2}{3^4}
\]
2. **Comparing with Students' Answers**:
- **Adam's answer**: \( \frac{2 \cdot 5^2}{3^4} \) - this is correct.
- **Nick's answer**: \( \frac{5^2}{2 \cdot 3^4} \) - this is incorrect; he moved the \( 2 \) to the denominator.
- **Shane's answer**: \( \frac{2 \cdot 3^4}{5^2} \) - this is incorrect; he moved \( 3^{-4} \) in the wrong direction and didn't simplify correctly.
3. **Correct Answer**:
The valid answer is Adam's. Therefore, the correct response would be:
**"Adam is correct because the negative exponent moves the base to the other side of the fraction bar and makes the exponent positive."**