Certainly! Let's walk through the problem step-by-step to understand how we arrive at the final simplified expression.
The expression we need to simplify is:
[tex]\[ \left(\frac{(x+y)^{2/3} \cdot (x-y)^{2/3}}{\sqrt{x+y} \cdot \sqrt{(x-y)^3}}\right)^6 \][/tex]
### Step 1: Simplify the Inner Expression
Let's first consider the inner expression inside the parentheses:
[tex]\[ \frac{(x+y)^{2/3} \cdot (x-y)^{2/3}}{\sqrt{x+y} \cdot \sqrt{(x-y)^3}} \][/tex]
#### Numerator:
The numerator is:
[tex]\[ (x+y)^{2/3} \cdot (x-y)^{2/3} \][/tex]
#### Denominator:
Let's simplify the denominator term by term:
[tex]\[ \sqrt{x+y} = (x+y)^{1/2} \][/tex]
[tex]\[ \sqrt{(x-y)^3} = ((x-y)^3)^{1/2} = (x-y)^{3/2} \][/tex]
So the denominator becomes:
[tex]\[ (x+y)^{1/2} \cdot (x-y)^{3/2} \][/tex]
### Step 2: Combine the Expressions
We now have:
[tex]\[ \frac{(x+y)^{2/3} \cdot (x-y)^{2/3}}{(x+y)^{1/2} \cdot (x-y)^{3/2}} \][/tex]
### Step 3: Apply the Exponent Properties
Let's handle the exponents of each term separately.
For \((x+y)\):
[tex]\[ \frac{(x+y)^{2/3}}{(x+y)^{1/2}} = (x+y)^{2/3 - 1/2} = (x+y)^{2/3 - 3/6} = (x+y)^{4/6 - 3/6} = (x+y)^{1/6} \][/tex]
For \((x-y)\):
[tex]\[ \frac{(x-y)^{2/3}}{(x-y)^{3/2}} = (x-y)^{2/3 - 3/2} = (x-y)^{2/3 - 9/6} = (x-y)^{4/6 - 9/6} = (x-y)^{-5/6} \][/tex]
Thus, the simplified form of the inner fraction is:
[tex]\[ (x+y)^{1/6} \cdot (x-y)^{-5/6} \][/tex]
### Step 4: Raise to the Power of 6
Now we raise the entire expression to the power of 6:
[tex]\[ \left((x+y)^{1/6} \cdot (x-y)^{-5/6}\right)^6 \][/tex]
Using the property of exponents \((a^m \cdot b^n)^p = a^{mp} \cdot b^{np}\):
[tex]\[ (x+y)^{1/6 \cdot 6} \cdot (x-y)^{-5/6 \cdot 6} = (x+y)^{1} \cdot (x-y)^{-5} \][/tex]
### Step 5: Write the Final Answer
The final simplified expression is:
[tex]\[ \frac{(x+y)^1}{(x-y)^5} = \frac{(x+y)}{(x-y)^5} \][/tex]
Thus, the simplified result is:
[tex]\[ \boxed{\frac{(x+y)}{(x-y)^5}} \][/tex]
