To simplify the given expression [tex]\(\frac{x y^{-6}}{x^{-4} y^2}\)[/tex], let's follow each step carefully:
1. Initial Expression:
[tex]\[
\frac{x y^{-6}}{x^{-4} y^2}
\][/tex]
2. Rewrite the negative exponents using properties of exponents:
- [tex]\(y^{-n} = \frac{1}{y^n}\)[/tex]
- [tex]\(x^{-m} = \frac{1}{x^m}\)[/tex]
Applying these properties, we get:
[tex]\[
x y^{-6} = x \cdot \frac{1}{y^6} = \frac{x}{y^6}
\][/tex]
and
[tex]\[
x^{-4} y^2 = \frac{1}{x^4} \cdot y^2 = \frac{y^2}{x^4}
\][/tex]
3. Rewrite the entire fraction:
[tex]\[
\frac{\frac{x}{y^6}}{\frac{y^2}{x^4}}
\][/tex]
4. Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator:
[tex]\[
\frac{x}{y^6} \times \frac{x^4}{y^2} = \frac{x \cdot x^4}{y^6 \cdot y^2} = \frac{x^{1+4}}{y^{6+2}} = \frac{x^5}{y^8}
\][/tex]
5. The final simplified expression:
[tex]\[
x^5 y^{-8}
\][/tex]
Therefore, after eliminating the negative exponents, the simplified result is:
[tex]\(\boxed{x^5 y^{-8}}\)[/tex].
