For a natural number n, let `alpha_(n) = 19^(n)-12^n` . Then, the value of `(31alpha_9-alpha_(10))/(57alpha_8)` is _______

Answer

To solve the problem, we need to find the value of \((31\alpha_9 - \alpha_{10}) / (57\alpha_8)\), where \(\alpha_n = 19^n - 12^n\).

### Step-by-Step Solution:

1. **Define \(\alpha_n\)**:
\[ \alpha_n = 19^n - 12^n \]

2. **Substitute for \(\alpha_9\), \(\alpha_{10}\), and \(\alpha_8\)**:
- \(\alpha_9 = 19^9 - 12^9\)
- \(\alpha_{10} = 19^{10} - 12^{10}\)
- \(\alpha_8 = 19^8 - 12^8\)

3. **Substitute into the expression**:
\[ 31\alpha_9 - \alpha_{10} = 31(19^9 - 12^9) - (19^{10} - 12^{10}) \]

4. **Expand the expression**:
\[ = 31 \cdot 19^9 - 31 \cdot 12^9 - 19^{10} + 12^{10} \]

5. **Rearrange the terms**:
\[ = (31 \cdot 19^9 - 19^{10}) + (12^{10} - 31 \cdot 12^9) \]
\[ = 19^9(31 - 19) + 12^9(12 - 31) \]
\[ = 19^9 \cdot 12 + 12^9 \cdot (-19) \]

6. **Factor out common terms**:
\[ = 19^9 \cdot 12 - 31 \cdot 12^9 \]

7. **Now substitute this back into the original expression**:
\[ \frac{31\alpha_9 - \alpha_{10}}{57\alpha_8} = \frac{19^9 \cdot 12 - 31 \cdot 12^9}{57(19^8 - 12^8)} \]

8. **Factor out \(12\) from the numerator**:
\[ = \frac{12(19^9 - 31 \cdot 12^8)}{57(19^8 - 12^8)} \]

9. **Recognize that \(19^9 - 31 \cdot 12^8 = 19^8(19 - 12) + 12^8(12 - 31)\)**:
\[ = \frac{12(19^8(19 - 12) + 12^8(12 - 31))}{57(19^8 - 12^8)} \]

10. **Notice that \(19 - 12 = 7\) and \(12 - 31 = -19\)**:
\[ = \frac{12(19^8 \cdot 7 - 12^8 \cdot 19)}{57(19^8 - 12^8)} \]

11. **Now, we can simplify the fraction**:
\[ = \frac{12 \cdot 7}{57} = \frac{84}{57} = \frac{4}{3} \]

Thus, the final value is:
\[ \frac{4}{3} \]