If `m = root(3)(15) and n = root(3)(14)`, find the value of `m - n - (1)/(m^(2) + mn + n^(2))`

Answer

To solve the problem, we need to find the value of \( m - n - \frac{1}{m^2 + mn + n^2} \) where \( m = \sqrt[3]{15} \) and \( n = \sqrt[3]{14} \).

### Step-by-Step Solution:

1. **Identify the values of \( m \) and \( n \)**:
\[ m = \sqrt[3]{15}, \quad n = \sqrt[3]{14} \]

2. **Calculate \( m - n \)**:
\[ m - n = \sqrt[3]{15} - \sqrt[3]{14} \]

3. **Rewrite the expression**:
We need to evaluate:
\[ m - n - \frac{1}{m^2 + mn + n^2} \]

4. **Calculate \( m^2 + mn + n^2 \)**:
Using the identity \( m^2 + mn + n^2 = (m-n)^2 + mn \):
- First, calculate \( m^2 \) and \( n^2 \):
\[ m^2 = (\sqrt[3]{15})^2 = \sqrt[3]{15^2} = \sqrt[3]{225} \]
\[ n^2 = (\sqrt[3]{14})^2 = \sqrt[3]{14^2} = \sqrt[3]{196} \]
- Now calculate \( mn \):
\[ mn = \sqrt[3]{15} \cdot \sqrt[3]{14} = \sqrt[3]{15 \cdot 14} = \sqrt[3]{210} \]
- Therefore:
\[ m^2 + mn + n^2 = \sqrt[3]{225} + \sqrt[3]{210} + \sqrt[3]{196} \]

5. **Use the identity for \( m^3 - n^3 \)**:
The difference of cubes can be expressed as:
\[ m^3 - n^3 = (m - n)(m^2 + mn + n^2) \]
- We know \( m^3 = 15 \) and \( n^3 = 14 \):
\[ m^3 - n^3 = 15 - 14 = 1 \]
- Thus:
\[ 1 = (m - n)(m^2 + mn + n^2) \]
- This implies:
\[ m^2 + mn + n^2 = \frac{1}{m - n} \]

6. **Substitute back into the expression**:
Now we can substitute \( m^2 + mn + n^2 \) back into the original expression:
\[ m - n - \frac{1}{m^2 + mn + n^2} = m - n - (m - n) = 0 \]

### Final Answer:
\[ \text{The value of } m - n - \frac{1}{m^2 + mn + n^2} \text{ is } 0. \]