If `agt0,bgt0,cgt0,` then the minimum value of
`sqrt((4a)/(b))+root(3)((27b)/(c))+root(4)((c)/(108a)),` is

Options

A. 4
B. 5
C. 3
D. 9

Explanation

To find the minimum value of the expression

\[ \sqrt{\frac{4a}{b}} + \sqrt[3]{\frac{27b}{c}} + \sqrt[4]{\frac{c}{108a}}, \]

we can apply the method of inequalities, specifically the weighted AM-GM inequality.

### Step-by-Step Solution:

1. **Identify the terms**: We have three terms in the expression:
- \( x_1 = \sqrt{\frac{4a}{b}} \)
- \( x_2 = \sqrt[3]{\frac{27b}{c}} \)
- \( x_3 = \sqrt[4]{\frac{c}{108a}} \)

2. **Assign weights**: We will assign weights to these terms based on their roots:
- Weight for \( x_1 \) is 2 (because it is a square root),
- Weight for \( x_2 \) is 3 (because it is a cube root),
- Weight for \( x_3 \) is 4 (because it is a fourth root).

3. **Apply the weighted AM-GM inequality**:
According to the weighted AM-GM inequality:
\[ \frac{L_1 x_1 + L_2 x_2 + L_3 x_3}{L_1 + L_2 + L_3} \geq (x_1^{L_1} x_2^{L_2} x_3^{L_3})^{\frac{1}{L_1 + L_2 + L_3}}, \]
where \( L_1 = 2, L_2 = 3, L_3 = 4 \).

4. **Calculate the total weights**:
\[ L_1 + L_2 + L_3 = 2 + 3 + 4 = 9. \]

5. **Substitute the terms into the inequality**:
\[ \frac{2\sqrt{\frac{4a}{b}} + 3\sqrt[3]{\frac{27b}{c}} + 4\sqrt[4]{\frac{c}{108a}}}{9} \geq \left(\sqrt{\frac{4a}{b}}^2 \cdot \sqrt[3]{\frac{27b}{c}}^3 \cdot \sqrt[4]{\frac{c}{108a}}^4\right)^{\frac{1}{9}}. \]

6. **Calculate the product**:
\[ = \left(\frac{4a}{b}\right)^{1} \cdot \left(\frac{27b}{c}\right)^{1} \cdot \left(\frac{c}{108a}\right)^{1} = \frac{4a \cdot 27b \cdot c}{b \cdot 108a} = \frac{108bc}{108} = c. \]

7. **Simplify the inequality**:
\[ \frac{2\sqrt{\frac{4a}{b}} + 3\sqrt[3]{\frac{27b}{c}} + 4\sqrt[4]{\frac{c}{108a}}}{9} \geq \left(c\right)^{\frac{1}{9}}. \]

8. **Multiply both sides by 9**:
\[ 2\sqrt{\frac{4a}{b}} + 3\sqrt[3]{\frac{27b}{c}} + 4\sqrt[4]{\frac{c}{108a}} \geq 9. \]

9. **Conclusion**: The minimum value of the expression is 9.

### Final Answer:
The minimum value of the expression is \( \boxed{9} \).