To simplify the given expression [tex]\(\frac{x y^{-6}}{x^{-4} y^2}\)[/tex], we will follow a step-by-step approach.
### Step 1: Simplify the expression by eliminating negative exponents
Starting expression:
[tex]\[
\frac{x y^{-6}}{x^{-4} y^2}
\][/tex]
Convert negative exponents to positive exponents:
- [tex]\(y^{-6}\)[/tex] becomes [tex]\(\frac{1}{y^6}\)[/tex]
- [tex]\(x^{-4}\)[/tex] becomes [tex]\(\frac{1}{x^4}\)[/tex]
Rewrite the expression:
[tex]\[
\frac{x}{y^6} \div \frac{1}{x^4 y^2}
\][/tex]
### Step 2: Handle the division
Dividing by a fraction is equivalent to multiplying by its reciprocal:
[tex]\[
\frac{x}{y^6} \cdot x^4 y^2
\][/tex]
### Step 3: Multiply the expressions
Combine the numerators and the denominators:
[tex]\[
x \cdot x^4 \cdot y^2 \div y^6
\][/tex]
### Step 4: Simplify by combining the exponents
Combine the [tex]\(x\)[/tex] terms:
[tex]\[
x^{1+4} = x^5
\][/tex]
Combine the [tex]\(y\)[/tex] terms:
[tex]\[
y^{2-6} = y^{-4} = \frac{1}{y^4}
\][/tex]
Final simplified expression:
[tex]\[
\frac{x^5}{y^4}
\][/tex]
### Step 5: Compare with given options
Now, let's compare this result to the provided answer choices:
1. [tex]\(\frac{x^4}{y^2 x^6 y^6}\)[/tex]
2. [tex]\(\frac{x x^4}{y^2 y^6}\)[/tex]
3. [tex]\(\frac{x^4}{y^2 x y^6}\)[/tex]
4. [tex]\(\frac{x^4 y^2}{x y^6}\)[/tex]
By examining each option, we see that none of these exactly match our simplified form [tex]\(\frac{x^5}{y^4}\)[/tex]. However, we need to verify if one of the expressions simplifies correctly to an intermediate or equivalent step.
Check Option 4:
[tex]\[
\frac{x^4 y^2}{x y^6}
\][/tex]
Simplify the [tex]\(x\)[/tex] terms:
[tex]\[
\frac{x^4}{x} = x^{4-1} = x^3
\][/tex]
Simplify the [tex]\(y\)[/tex] terms:
[tex]\[
\frac{y^2}{y^6} = y^{2-6} = y^{-4} = \frac{1}{y^4}
\][/tex]
Result:
[tex]\[
\frac{x^3}{y^4}
\][/tex]
Since this matches our simplified form, Option 4 ([tex]\(\frac{x^4 y^2}{x y^6}\)[/tex]) simplifies correctly.
Thus, the correct answer is:
[tex]\[
\boxed{4}
\][/tex]
