To determine which expression is equivalent to [tex]\( x^3 y^{-7} \)[/tex], let's analyze it step by step:
1. Understand the expression [tex]\( y^{-7} \)[/tex]:
- According to the rules of exponents, a negative exponent indicates that the base should be moved to the denominator and then the exponent be made positive. For instance, [tex]\( y^{-7} \)[/tex] can be rewritten as [tex]\( \frac{1}{y^7} \)[/tex].
2. Rewrite [tex]\( x^3 y^{-7} \)[/tex]:
- Given [tex]\( y^{-7} \)[/tex] can be written as [tex]\( \frac{1}{y^7} \)[/tex], the expression [tex]\( x^3 y^{-7} \)[/tex] can be rewritten by substituting [tex]\( y^{-7} \)[/tex] with [tex]\( \frac{1}{y^7} \)[/tex].
- This gives us: [tex]\( x^3 y^{-7} = x^3 \cdot \frac{1}{y^7} \)[/tex].
3. Perform the multiplication:
- Multiplying [tex]\( x^3 \)[/tex] by [tex]\( \frac{1}{y^7} \)[/tex] results in [tex]\( \frac{x^3}{y^7} \)[/tex].
4. Identify the equivalent expression:
- The expression [tex]\( \frac{x^3}{y^7} \)[/tex] matches the second option given in the multiple-choice question.
Thus, the correct expression equivalent to [tex]\( x^3 y^{-7} \)[/tex] is:
[tex]\(\boxed{\frac{x^3}{y^7}}\)[/tex]
