`(a+b)(sqrt(a) + sqrt(b))(sqrt(a)- sqrt(b))` = ______

Options

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

Answer

To solve the expression \((a+b)(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b})\), we can follow these steps:

### Step 1: Identify the Expression
We start with the expression:
\[ (a+b)(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) \]

### Step 2: Recognize the Identity
Notice that \((\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b})\) is a difference of squares. We can use the identity:
\[ (x + y)(x - y) = x^2 - y^2 \]
Here, let \(x = \sqrt{a}\) and \(y = \sqrt{b}\).

### Step 3: Apply the Identity
Using the identity, we can simplify:
\[ (\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = (\sqrt{a})^2 - (\sqrt{b})^2 = a - b \]

### Step 4: Substitute Back
Now substitute back into the original expression:
\[ (a+b)(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = (a+b)(a-b) \]

### Step 5: Expand the Expression
Now, we expand \((a+b)(a-b)\) using the distributive property:
\[ (a+b)(a-b) = a^2 - ab + ba - b^2 = a^2 - b^2 \]

### Final Result
Thus, the final simplified expression is:
\[ \boxed{a^2 - b^2} \]

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