To simplify the given expression [tex]\(\frac{x \cdot y^{-6}}{x^{-4} \cdot y^2}\)[/tex] and eliminate the negative exponents, we should carefully follow these steps:
1. Rewrite negative exponents as positive exponents using their reciprocal property:
The expression [tex]\(\frac{x \cdot y^{-6}}{x^{-4} \cdot y^2}\)[/tex] can be rewritten by moving the terms with negative exponents to the other part of the fraction. So, [tex]\(y^{-6}\)[/tex] in the numerator moves to the denominator as [tex]\(y^6\)[/tex], and [tex]\(x^{-4}\)[/tex] in the denominator moves to the numerator as [tex]\(x^4\)[/tex]. This gives us:
[tex]\[
\frac{x \cdot x^4}{y^2 \cdot y^6}
\][/tex]
2. Combine the terms with the same base in the numerator and the denominator:
- Combine [tex]\(x\)[/tex] and [tex]\(x^4\)[/tex]:
[tex]\[
x \cdot x^4 = x^{1+4} = x^5
\][/tex]
- Combine [tex]\(y^2\)[/tex] and [tex]\(y^6\)[/tex]:
[tex]\[
y^2 \cdot y^6 = y^{2+6} = y^8
\][/tex]
3. Construct the simplified expression:
Therefore, the simplified expression is:
[tex]\[
\frac{x^5}{y^8}
\][/tex]
Upon reviewing the provided options, none of these directly match [tex]\(\frac{x^5}{y^8}\)[/tex]. Hence, the simplified expression equal to [tex]\(\frac{x^5}{y^8}\)[/tex] is not available among the listed options.
Therefore, the correct answer is:
- None of the provided options
