Simplify :
` : ( ( x+(1)/(y)) ^(a) ( x-(1)/(y))^(b))/((y+(1)/(x))^(a) (y-(1)/(x))^(b)) `

Answer

To simplify the expression

\[ \frac{(x + \frac{1}{y})^a (x - \frac{1}{y})^b}{(y + \frac{1}{x})^a (y - \frac{1}{x})^b} \]

we will follow these steps:

### Step 1: Rewrite the expression
Rewrite the expression to make it clearer:

\[ \frac{(x + \frac{1}{y})^a (x - \frac{1}{y})^b}{(y + \frac{1}{x})^a (y - \frac{1}{x})^b} \]

### Step 2: Find a common denominator
For the terms in the numerator and denominator, we can find a common denominator:

The numerator becomes:

\[ (xy + 1)^a (xy - 1)^b \]

The denominator becomes:

\[ (xy + 1)^b (xy - 1)^a \]

### Step 3: Simplify the expression
Now we can simplify the expression:

\[ \frac{(xy + 1)^a (xy - 1)^b}{(xy + 1)^b (xy - 1)^a} \]

This can be simplified to:

\[ \frac{(xy + 1)^{a-b} (xy - 1)^{b-a}} \]

### Step 4: Final simplification
We can express this as:

\[ \frac{(xy + 1)^{a-b}}{(xy - 1)^{a-b}} \]

### Step 5: Final result
Thus, the simplified expression is:

\[ \left(\frac{xy + 1}{xy - 1}\right)^{a-b} \]