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. 5.375
B. 1
C. 6
D. 5

Answer

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

### Step 1: Define Variables
Let:
- \( A = \sqrt[3]{3.5} \)
- \( B = \sqrt[3]{2.5} \)

### Step 2: Rewrite the Expression
The expression can be rewritten using the variables:
\[ (A + B) \left( A^2 - \sqrt[3]{8.75} + B^2 \right) \]

### Step 3: Simplify \( \sqrt[3]{8.75} \)
We can express \( 8.75 \) in terms of \( A \) and \( B \):
\[ 8.75 = 3.5 \times 2.5 = (A^3)(B^3) \]
Thus, we have:
\[ \sqrt[3]{8.75} = \sqrt[3]{3.5 \times 2.5} = AB \]

### Step 4: Substitute Back into the Expression
Now, we can substitute back into the expression:
\[ (A + B) \left( A^2 - AB + B^2 \right) \]

### Step 5: Recognize the Identity
The expression \( A^2 - AB + B^2 \) can be recognized as:
\[ A^2 - AB + B^2 = \frac{(A + B)^3 - (A^3 + B^3)}{A + B} \]
However, we can also use the identity for the sum of cubes:
\[ A^3 + B^3 = (A + B)(A^2 - AB + B^2) \]

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

### Step 7: Substitute Back to Find the Final Value
Now, substituting back, we have:
\[ (A + B)(A^2 - AB + B^2) = A^3 + B^3 = 6 \]

### Conclusion
The value of the original expression is \( 6 \).