Let's carefully analyze Yenifer's steps to simplify the given expression and identify which parts of her answer are incorrect.
### Step-by-Step Simplification:
1. Original Problem:
[tex]\[
\left(\frac{4 x^{-2} y^2}{2 x^2 y^7}\right)^3
\][/tex]
2. Simplify Inside the Parentheses:
[tex]\[
\frac{4 x^{-2} y^2}{2 x^2 y^7}
\][/tex]
- Coefficient: [tex]\(\frac{4}{2} = 2\)[/tex]
- [tex]\(x\)[/tex] term: [tex]\(x^{-2} \cdot x^{-2} = x^{-4}\)[/tex]
- [tex]\(y\)[/tex] term: [tex]\(y^2 \cdot y^{-7} = y^{-5}\)[/tex]
So, the expression simplifies to:
[tex]\[
2 x^{-4} y^{-5}
\][/tex]
3. Raise Each Term to the Power of 3:
[tex]\[
(2 x^{-4} y^{-5})^3
\][/tex]
- Coefficient: [tex]\(2^3 = 8\)[/tex]
- [tex]\(x\)[/tex] term: [tex]\((x^{-4})^3 = x^{-12}\)[/tex]
- [tex]\(y\)[/tex] term: [tex]\((y^{-5})^3 = y^{-15}\)[/tex]
Thus, the expression becomes:
[tex]\[
8 x^{-12} y^{-15}
\][/tex]
4. Write the expression in the correct format (with positive exponents):
[tex]\[
8 \cdot \frac{1}{x^{12} y^{15}} = \frac{8}{x^{12} y^{15}}
\][/tex]
5. Compare with Yenifer's Answer:
Yenifer's simplified result is:
[tex]\[
\frac{8}{x y^3}
\][/tex]
By comparing each part of Yenifer's answer with the correctly simplified result:
- Coefficient: Both results have the coefficient 8, which is correct.
- [tex]\(x\)[/tex] Exponent: Yenifer's answer has [tex]\(x^1\)[/tex] (or simply x) in the denominator, while the correct exponent should be [tex]\(x^{12}\)[/tex].
- [tex]\(y\)[/tex] Exponent: Yenifer's answer has [tex]\(y^3\)[/tex] in the denominator, while the correct exponent should be [tex]\(y^{15}\)[/tex].
### Incorrect Parts:
- The x exponent in Yenifer's answer is incorrect.
- The y exponent in Yenifer's answer is incorrect.
### Conclusion:
Therefore, the parts of Yenifer's answer that are incorrect are:
- II. The x exponent
- III. The y exponent
Thus, the correct answer is:
[tex]\[
\boxed{\text{II and III}}
\][/tex]
