Solve the following :
`[(x + 1/y)^a (x - 1/y)^b] div [(y + 1/x)^a (y - 1/x)^b]` is equal to

Options

A. `(x/y)^(a + b)`
B. `(y/x)^(a + b)`
C. `x^a/y^b`
D. `(xy)^(a + b)`

Answer

To solve 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 simplify it step by step.

### Step 1: Rewrite the expression
We start with the given expression:

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

### Step 2: Factor the terms
Notice that we can factor the numerator and the denominator separately.

For the numerator:
- \( (x + \frac{1}{y})^a \) can be rewritten as \( \left(\frac{xy + 1}{y}\right)^a \)
- \( (x - \frac{1}{y})^b \) can be rewritten as \( \left(\frac{xy - 1}{y}\right)^b \)

Thus, the numerator becomes:

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

For the denominator:
- \( (y + \frac{1}{x})^a \) can be rewritten as \( \left(\frac{yx + 1}{x}\right)^a \)
- \( (y - \frac{1}{x})^b \) can be rewritten as \( \left(\frac{yx - 1}{x}\right)^b \)

Thus, the denominator becomes:

\[ \frac{(yx + 1)^a}{x^a} \cdot \frac{(yx - 1)^b}{x^b} = \frac{(yx + 1)^a (yx - 1)^b}{x^{a+b}} \]

### Step 3: Combine the fractions
Now, we can rewrite the entire expression as:

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

This can be expressed as:

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

### Step 4: Simplify the expression
Notice that \( xy = yx \), thus the terms \( (xy + 1)^a \) and \( (yx + 1)^a \) will cancel out, as will \( (xy - 1)^b \) and \( (yx - 1)^b \). Therefore, we are left with:

\[ \frac{x^{a+b}}{y^{a+b}} = \left(\frac{x}{y}\right)^{a+b} \]

### Final Result
Thus, the simplified expression is:

\[ \left(\frac{x}{y}\right)^{a+b} \]