Which shows the following expression after the negative exponents have been eliminated?

[tex]\[ \frac{x y^{-6}}{x^{-4} y^2}, \quad x \neq 0, \quad y \neq 0 \][/tex]

A. [tex]\(\frac{x^4}{y^2 x^6 y^6}\)[/tex]

B. [tex]\(\frac{x x^4}{y^2 y^6}\)[/tex]

C. [tex]\(\frac{x^4}{y^2 x y^6}\)[/tex]

D. [tex]\(\frac{x^4 y^2}{x y^6}\)[/tex]



Answer :

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].