To simplify the given expression [tex]\(\frac{x y^{-6}}{x^{-4} y^2}\)[/tex] and eliminate the negative exponents, let's break it down step-by-step:
1. Rewrite the negative exponents as positive exponents:
[tex]\[
x y^{-6} = \frac{x}{y^6}
\][/tex]
[tex]\[
x^{-4} y^2 = \frac{y^2}{x^4}
\][/tex]
So, the expression [tex]\(\frac{x y^{-6}}{x^{-4} y^2}\)[/tex] becomes:
[tex]\[
\frac{\frac{x}{y^6}}{\frac{y^2}{x^4}}
\][/tex]
2. Simplify the complex fraction:
To simplify [tex]\(\frac{\frac{x}{y^6}}{\frac{y^2}{x^4}}\)[/tex], multiply the numerator by the reciprocal of the denominator:
[tex]\[
\frac{x}{y^6} \div \frac{y^2}{x^4} = \frac{x}{y^6} \cdot \frac{x^4}{y^2}
\][/tex]
3. Combine the fractions:
Multiply the numerators together and the denominators together:
[tex]\[
\frac{x \cdot x^4}{y^6 \cdot y^2} = \frac{x^{1+4}}{y^{6+2}} = \frac{x^5}{y^8}
\][/tex]
Now, we need to match this simplified expression [tex]\(\frac{x^5}{y^8}\)[/tex] with one of the provided options.
Let's evaluate each option:
- [tex]\(\frac{x^4}{y^2 x^6 y^6}\)[/tex]:
[tex]\[
= \frac{x^4}{x^6 y^8} = \frac{1}{x^2 y^8}
\][/tex]
- [tex]\(\frac{x x^4}{y^2 y^6}\)[/tex]:
[tex]\[
= \frac{x^5}{y^8}
\][/tex]
- [tex]\(\frac{x^4}{y^2 x y^6}\)[/tex]:
[tex]\[
= \frac{x^4}{x y^8} = \frac{x^3}{y^8}
\][/tex]
- [tex]\(\frac{x^4 y^2}{x y^6}\)[/tex]:
[tex]\[
= \frac{x^3 y^2}{y^6} = \frac{x^3}{y^4}
\][/tex]
The correct option that matches [tex]\(\frac{x^5}{y^8}\)[/tex] is:
[tex]\[
\boxed{\frac{x x^4}{y^2 y^6}}
\][/tex]
