Let us simplify the given expression:
[tex]\[
(x y)^{\frac{1}{x-y}}
\][/tex]
We can utilize the properties of exponents to simplify this expression. The key property we'll use is:
[tex]\[
(ab)^c = a^c \cdot b^c
\][/tex]
This property states that raising a product to a power is the same as raising each factor to that power separately and then multiplying the results.
Applying this property to our expression, we have:
[tex]\[
(x y)^{\frac{1}{x-y}} = x^{\frac{1}{x-y}} \cdot y^{\frac{1}{x-y}}
\][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[
(x y)^{\frac{1}{x-y}} = x^{\frac{1}{x-y}} \cdot y^{\frac{1}{x-y}}
\][/tex]
Thus, our final answer is:
[tex]\[
x^{\frac{1}{x-y}} \cdot y^{\frac{1}{x-y}}
\][/tex]
