If `(a^(3//2)-ab^(1//2)+a^(1//2)b-b^(3//2))` is divided by `(a^(1//2)-b^(1//2))`, then the quotient is :

Options

A. `a+b`
B. `a-b`
C. `a^(1//2)+b^(1//2)`
D. `a^(2)-b^(2)`

Answer

To solve the problem, we need to divide the expression \( a^{3/2} - ab^{1/2} + a^{1/2}b - b^{3/2} \) by \( a^{1/2} - b^{1/2} \). Let's go through the steps systematically.

### Step 1: Rewrite the expression
We start with the expression:
\[ a^{3/2} - ab^{1/2} + a^{1/2}b - b^{3/2} \]

### Step 2: Group the terms
We can rearrange the expression to group similar terms:
\[ (a^{3/2} - b^{3/2}) + (a^{1/2}b - ab^{1/2}) \]

### Step 3: Factor the grouped terms
Now we can factor each group:
1. The first group \( a^{3/2} - b^{3/2} \) can be factored using the difference of cubes:
\[ a^{3/2} - b^{3/2} = (a^{1/2} - b^{1/2})(a + b^{1/2}) \]
2. The second group \( a^{1/2}b - ab^{1/2} \) can be factored as:
\[ a^{1/2}b - ab^{1/2} = (a^{1/2} - b^{1/2})ab^{1/2} \]

### Step 4: Combine the factors
Now we can combine the factored forms:
\[ (a^{1/2} - b^{1/2})(a + b^{1/2}) + (a^{1/2} - b^{1/2})ab^{1/2} \]
Factoring out \( (a^{1/2} - b^{1/2}) \):
\[ (a^{1/2} - b^{1/2}) \left( (a + b^{1/2}) + ab^{1/2} \right) \]

### Step 5: Simplify the expression
Now we simplify the expression inside the parentheses:
\[ (a + b^{1/2}) + ab^{1/2} = a + b^{1/2} + ab^{1/2} \]

### Step 6: Divide by \( a^{1/2} - b^{1/2} \)
Now we divide the entire expression by \( a^{1/2} - b^{1/2} \):
\[ \frac{(a^{1/2} - b^{1/2}) (a + b^{1/2} + ab^{1/2})}{a^{1/2} - b^{1/2}} \]
The \( a^{1/2} - b^{1/2} \) cancels out:
\[ a + b^{1/2} + ab^{1/2} \]

### Step 7: Final Quotient
Thus, the final quotient is:
\[ a + b \]

### Conclusion
The quotient when \( a^{3/2} - ab^{1/2} + a^{1/2}b - b^{3/2} \) is divided by \( a^{1/2} - b^{1/2} \) is \( a + b \).