Alright, let's break down the expression [tex]\((xy)^{\frac{1}{(x-y)}}\)[/tex] step by step.
To solve this expression, we need to understand a few mathematical concepts, including exponents and the properties of powers.
#### Step 1: Understand the Expression
The expression we need to solve is [tex]\((xy)^{\frac{1}{(x-y)}}\)[/tex].
This expression represents a number [tex]\(xy\)[/tex] raised to the power of [tex]\(\frac{1}{(x-y)}\)[/tex].
#### Step 2: Identify Variables
We have two variables here:
- [tex]\(x\)[/tex]: This is one of the numbers in multiplication.
- [tex]\(y\)[/tex]: This is the second number in multiplication.
#### Step 3: Exponentiation
The form [tex]\((xy)^{\frac{1}{(x-y)}}\)[/tex] can be interpreted as a power or a root. Specifically, when [tex]\(xy\)[/tex] is raised to the power [tex]\(\frac{1}{(x-y)}\)[/tex], it is equivalent to finding the [tex]\((x-y)\)[/tex]-th root of [tex]\(xy\)[/tex].
#### Step 4: Substitution
To simplify the explanation, let's consider an example. Suppose we know some values for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
- Let [tex]\(x = 4\)[/tex]
- Let [tex]\(y = 2\)[/tex]
Substituting these values into the expression:
[tex]\[
(xy)^{\frac{1}{(x-y)}} = (4 \cdot 2)^{\frac{1}{(4-2)}}
\][/tex]
Calculate the multiplication inside the parentheses first:
[tex]\[
(4 \cdot 2) = 8
\][/tex]
Next, we evaluate the exponent:
[tex]\[
\frac{1}{(4-2)} = \frac{1}{2}
\][/tex]
So, the expression now is:
[tex]\[
8^{\frac{1}{2}}
\][/tex]
#### Step 5: Calculate the Final Result
[tex]\[
8^{\frac{1}{2}} \text{ is the square root of } 8.
\][/tex]
Thus:
[tex]\[
8^{\frac{1}{2}} = \sqrt{8} \approx 2.828
\][/tex]
#### Conclusion
Hence, after evaluating the expression step by step, we determined that:
[tex]\[
(x y)^{\frac{1}{(x-y)}} \equiv \sqrt{8} \approx 2.828 \text{ when } x = 4 \text{ and } y = 2.
\][/tex]
Always remember that this specific kind of exponentiation determines the radical or root according to the denominator of the exponent.
