The value of `(root(3)(3.5)+root(3)(2.5)){(root(3)(3.5))^(2)-root(3)(8.75)+(root(3)(2.5))^(2)}` is :

Options

A. A) `5.375`
B. B) `1`
C. C) `6`
D. D) `5`

Answer

To solve the expression \((\sqrt[3]{3.5} + \sqrt[3]{2.5}) \cdot \left((\sqrt[3]{3.5})^2 - \sqrt[3]{8.75} + (\sqrt[3]{2.5})^2\right)\), we can follow these steps:

### Step 1: Simplify the expression
We can denote:
- Let \( A = \sqrt[3]{3.5} \)
- Let \( B = \sqrt[3]{2.5} \)

Thus, the expression can be rewritten as:
\[ (A + B) \cdot (A^2 - \sqrt[3]{8.75} + B^2) \]

### Step 2: Simplify \(\sqrt[3]{8.75}\)
Notice that:
\[ 8.75 = 3.5 \times 2.5 \]
Therefore:
\[ \sqrt[3]{8.75} = \sqrt[3]{3.5 \times 2.5} = \sqrt[3]{3.5} \cdot \sqrt[3]{2.5} = A \cdot B \]

### Step 3: Substitute back into the expression
Now, we can substitute \(\sqrt[3]{8.75}\) back into the expression:
\[ (A + B) \cdot (A^2 - A \cdot B + B^2) \]

### Step 4: Recognize the identity
We can use the identity for the sum of cubes:
\[ A^3 + B^3 = (A + B)(A^2 - AB + B^2) \]
Thus:
\[ A^2 - AB + B^2 = \frac{A^3 + B^3}{A + B} \]

### Step 5: Calculate \(A^3 + B^3\)
Now we calculate \(A^3\) and \(B^3\):
\[ A^3 = 3.5 \quad \text{and} \quad B^3 = 2.5 \]
So:
\[ A^3 + B^3 = 3.5 + 2.5 = 6 \]

### Step 6: Substitute back into the identity
Now substituting back:
\[ A + B = \sqrt[3]{3.5} + \sqrt[3]{2.5} \]
Thus:
\[ (A + B) \cdot (A^2 - AB + B^2) = (A + B) \cdot \frac{6}{A + B} = 6 \]

### Final Answer
The value of the expression is:
\[ \boxed{6} \]