Sure, let's simplify the expression [tex]\((xy)^{\frac{1}{x-y}}\)[/tex].
1. Understand the Base and Exponent: The given expression is [tex]\((xy)^{\frac{1}{x-y}}\)[/tex], where [tex]\(xy\)[/tex] is the base and [tex]\(\frac{1}{x-y}\)[/tex] is the exponent.
2. Properties of Exponents: Recall the properties of exponents. One relevant property is that [tex]\((a \cdot b)^c = a^c \cdot b^c\)[/tex]. However, in this problem, due to the nature of the variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex] in the exponent, this property doesn't directly apply for simplification.
3. Examine the Expression: There are no factors or terms that can be directly canceled or simplified further given the nature of the base [tex]\(xy\)[/tex] and the complex exponent [tex]\(\frac{1}{x-y}\)[/tex].
4. Consider Functional Dependence: The expression [tex]\((xy)^{\frac{1}{x-y}}\)[/tex] remains in its given form because the exponent [tex]\(\frac{1}{x-y}\)[/tex] can't be simplified further without additional context or values for [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
Therefore, the simplified form of [tex]\((xy)^{\frac{1}{x-y}}\)[/tex] is:
[tex]\[
(xy)^{\frac{1}{x-y}}
\][/tex]
Based on mathematical principles and understanding of exponents and algebraic expressions, the result of the simplification is:
[tex]\[
(xy)^{\frac{1}{x-y}}
\][/tex]
This shows the expression is already in its simplest form under the given conditions.
