To start solving the given expression and simplifying it, we need to follow these steps:
[tex]\[
\frac{x y^{-6}}{x^{-4} y^2}
\][/tex]
1. Dealing with Negative Exponents:
The negative exponents in the expression can be flipped to the other part of the fraction. Applying this property, [tex]\( a^{-b} = \frac{1}{a^b} \)[/tex]:
[tex]\[
\frac{x}{y^6} \div \frac{1}{x^4 \cdot y^2}
\][/tex]
The division by a fraction is equivalent to multiplying by its reciprocal. Therefore:
[tex]\[
\frac{x}{y^6} \times \frac{x^4 \cdot y^2}{1}
\][/tex]
2. Simplifying the Fractions:
Now you multiply the numerators together and the denominators together:
[tex]\[
\frac{x \cdot x^4 \cdot y^2}{y^6 \cdot 1}
\][/tex]
3. Combining Like Terms:
When multiplying terms with the same base, you add their exponents. So:
[tex]\[
x^{1+4} = x^5 \quad \text{and} \quad y^{2+6} = y^8
\][/tex]
Therefore:
[tex]\[
\frac{x^5}{y^8}
\][/tex]
This is the simplified form of the given expression after eliminating the negative exponents. The correct simplified form of the given expression is thus:
[tex]\[
\frac{x^5}{y^8}
\][/tex]
Therefore, the final simplified expression shows:
[tex]\[
\frac{x^5}{y^8}
\][/tex]
This represents the original mathematical operation with negative exponents eliminated and simplified correctly.
