`(root3(3)+root3(2))(root3(9)+root3(4)-root3(6))=`

Options

A. 5
B. `root9(5)`
C. `root6(5)`
D. `root3(5)`

Answer

To solve the expression \((\sqrt[3]{3} + \sqrt[3]{2})(\sqrt[3]{9} + \sqrt[3]{4} - \sqrt[3]{6})\), we will follow these steps:

### Step 1: Expand the expression
We will use the distributive property (also known as the FOIL method for binomials) to expand the expression:

\[ (\sqrt[3]{3} + \sqrt[3]{2})(\sqrt[3]{9} + \sqrt[3]{4} - \sqrt[3]{6}) = \sqrt[3]{3} \cdot \sqrt[3]{9} + \sqrt[3]{3} \cdot \sqrt[3]{4} - \sqrt[3]{3} \cdot \sqrt[3]{6} + \sqrt[3]{2} \cdot \sqrt[3]{9} + \sqrt[3]{2} \cdot \sqrt[3]{4} - \sqrt[3]{2} \cdot \sqrt[3]{6} \]

### Step 2: Simplify each term
Now, we will simplify each term:

1. \(\sqrt[3]{3} \cdot \sqrt[3]{9} = \sqrt[3]{27} = 3\)
2. \(\sqrt[3]{3} \cdot \sqrt[3]{4} = \sqrt[3]{12}\)
3. \(-\sqrt[3]{3} \cdot \sqrt[3]{6} = -\sqrt[3]{18}\)
4. \(\sqrt[3]{2} \cdot \sqrt[3]{9} = \sqrt[3]{18}\)
5. \(\sqrt[3]{2} \cdot \sqrt[3]{4} = \sqrt[3]{8} = 2\)
6. \(-\sqrt[3]{2} \cdot \sqrt[3]{6} = -\sqrt[3]{12}\)

So, we can rewrite the expression as:

\[ 3 + \sqrt[3]{12} - \sqrt[3]{18} + \sqrt[3]{18} + 2 - \sqrt[3]{12} \]

### Step 3: Combine like terms
Now, we combine the like terms:

\[ 3 + 2 + (\sqrt[3]{12} - \sqrt[3]{12}) + (\sqrt[3]{18} - \sqrt[3]{18}) = 5 \]

### Final Answer
Thus, the final answer is:

\[ \boxed{5} \]