\( ( x y ^ { 10 } \cdot x ^ { - 8 } \cdot y ^ { - 12 } \cdot y ) : ^ { \prime \prime } ( \frac { y ^ { - 4 } } { x ^ { - 8 } } \)

To simplify the expression \( ( x y^{10} \cdot x^{-8} \cdot y^{-12} \cdot y) : \left( \frac{y^{-4}}{x^{-8}} \right) \), we will first simplify the numerator and the denominator separately. ### Step 1: Simplify the numerator The numerator is: \[ x y^{10} \cdot x^{-8} \cdot y^{-12} \cdot y \] We can combine the terms involving \(x\) and \(y\): 1. For \(x\): \[ x \cdot x^{-8} = x^{1 - 8} = x^{-7} \] 2. For \(y\): \[ y^{10} \cdot y^{-12} \cdot y = y^{10 - 12 + 1} = y^{-1} \] So, the numerator simplifies to: \[ x^{-7} y^{-1} \] ### Step 2: Simplify the denominator The denominator is: \[ \frac{y^{-4}}{x^{-8}} \] This can be rewritten as: \[ y^{-4} \cdot x^{8} \] ### Step 3: Combine the numerator and denominator Now we can rewrite the entire expression: \[ \frac{x^{-7} y^{-1}}{y^{-4} \cdot x^{8}} = \frac{x^{-7} y^{-1}}{x^{8} y^{-4}} \] This can be simplified by separating the \(x\) and \(y\) terms: \[ = \frac{x^{-7}}{x^{8}} \cdot \frac{y^{-1}}{y^{-4}} \] ### Step 4: Simplify each part 1. For \(x\): \[ \frac{x^{-7}}{x^{8}} = x^{-7 - 8} = x^{-15} \] 2. For \(y\): \[ \frac{y^{-1}}{y^{-4}} = y^{-1 - (-4)} = y^{-1 + 4} = y^{3} \] ### Final Result Combining these results, we have: \[ x^{-15} y^{3} \] This can also be expressed as: \[ \frac{y^{3}}{x^{15}} \] Thus, the final simplified expression is: \[ \frac{y^{3}}{x^{15}} \]