Suppose a, b, c `in R, a ne 0` .If one root of `ax^(2) + bx + c = 0` is the fourth power of the other, then `(a^(4) c) ^(1//5) + (ac^(4)) ^(1//5) + b` = _______

Answer

To solve the problem, we need to analyze the given quadratic equation and the relationship between its roots.

### Step-by-step Solution:

1. **Identify the Roots**:
Let the roots of the quadratic equation \( ax^2 + bx + c = 0 \) be \( \alpha \) and \( \alpha^4 \), where \( \alpha^4 \) is the fourth power of \( \alpha \).

2. **Use Vieta's Formulas**:
According to Vieta's formulas:
- The sum of the roots \( \alpha + \alpha^4 = -\frac{b}{a} \)
- The product of the roots \( \alpha \cdot \alpha^4 = \alpha^5 = \frac{c}{a} \)

3. **Express \( b \) in terms of \( \alpha \)**:
From the sum of the roots:
\[ \alpha + \alpha^4 = -\frac{b}{a} \implies b = -a(\alpha + \alpha^4) \]

4. **Express \( c \) in terms of \( \alpha \)**:
From the product of the roots:
\[ \alpha^5 = \frac{c}{a} \implies c = a\alpha^5 \]

5. **Substitute \( c \) into the expression**:
We need to evaluate:
\[ (a^4 c)^{1/5} + (a c^4)^{1/5} + b \]
Substitute \( c = a\alpha^5 \):
\[ (a^4 (a\alpha^5))^{1/5} + (a (a\alpha^5)^4)^{1/5} + b \]

6. **Calculate \( (a^4 c)^{1/5} \)**:
\[ (a^5 \alpha^5)^{1/5} = a^{1} \alpha \]

7. **Calculate \( (a c^4)^{1/5} \)**:
First, find \( c^4 \):
\[ c^4 = (a\alpha^5)^4 = a^4 \alpha^{20} \]
Thus,
\[ (a (a^4 \alpha^{20}))^{1/5} = (a^5 \alpha^{20})^{1/5} = a^{1} \alpha^{4} \]

8. **Combine the results**:
Now we have:
\[ a \alpha + a \alpha^4 + b \]
Substitute \( b = -a(\alpha + \alpha^4) \):
\[ a \alpha + a \alpha^4 - a(\alpha + \alpha^4) = 0 \]

9. **Final Result**:
Therefore, the expression evaluates to:
\[ (a^4 c)^{1/5} + (a c^4)^{1/5} + b = 0 \]

### Final Answer:
\[ \boxed{0} \]