If `(x ^( 3//2) - xy ^(1//2)+ x ^(1//2) y - y ^(3//2))` is divided by `(x ^( 1//2) - y ^(1//2)),` the quotient is :

Options

A. `x + y `
B. `x -y`
C. `x ^(1//2) + y ^(1//2)`
D. `x ^(2) - y ^(2)`

Answer

To solve the problem of dividing the expression \( x^{3/2} - xy^{1/2} + x^{1/2}y - y^{3/2} \) by \( x^{1/2} - y^{1/2} \), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ x^{3/2} - xy^{1/2} + x^{1/2}y - y^{3/2} \]

### Step 2: Group the terms
We can group the terms in pairs:
\[ (x^{3/2} - xy^{1/2}) + (x^{1/2}y - y^{3/2}) \]

### Step 3: Factor out common terms
Now, we will factor out common terms from each group:
- From the first group \( x^{3/2} - xy^{1/2} \), we can factor out \( x^{1/2} \):
\[ x^{1/2}(x - y^{1/2}) \]
- From the second group \( x^{1/2}y - y^{3/2} \), we can factor out \( y^{1/2} \):
\[ y^{1/2}(x^{1/2} - y) \]

### Step 4: Combine the factored terms
Combining the factored terms, we have:
\[ x^{1/2}(x - y^{1/2}) + y^{1/2}(x^{1/2} - y) \]

### Step 5: Factor by grouping
Notice that we can factor out \( (x^{1/2} - y^{1/2}) \):
\[ (x^{1/2} + y)(x^{1/2} - y^{1/2}) \]

### Step 6: Write the complete expression
Thus, the complete expression can be rewritten as:
\[ (x^{1/2} + y)(x^{1/2} - y^{1/2}) \]

### Step 7: Divide by \( x^{1/2} - y^{1/2} \)
Now, we divide the entire expression by \( x^{1/2} - y^{1/2} \):
\[ \frac{(x^{1/2} + y)(x^{1/2} - y^{1/2})}{x^{1/2} - y^{1/2}} \]

### Step 8: Simplify the expression
The \( (x^{1/2} - y^{1/2}) \) terms cancel out:
\[ x^{1/2} + y \]

### Final Answer
Thus, the quotient is:
\[ \boxed{x^{1/2} + y} \]