If (3x - 2y) : (2x + 3y) = 5 : 6 then the value of `((root(3)(x)+root(3)(y))/(root(3)(x)-root(3)(y)))` is-

Options

A. `1/25`
B. 5
C. `1/5`
D. 25

Answer

To solve the problem, we need to find the value of the expression \(\frac{\sqrt[3]{x} + \sqrt[3]{y}}{\sqrt[3]{x} - \sqrt[3]{y}}\) given that \((3x - 2y) : (2x + 3y) = 5 : 6\).

### Step-by-Step Solution

1. **Set up the equation from the ratio**:
\[ \frac{3x - 2y}{2x + 3y} = \frac{5}{6} \]

2. **Cross-multiply to eliminate the fraction**:
\[ 6(3x - 2y) = 5(2x + 3y) \]

3. **Distribute both sides**:
\[ 18x - 12y = 10x + 15y \]

4. **Rearrange the equation**:
- Move all terms involving \(x\) to one side and all terms involving \(y\) to the other side:
\[ 18x - 10x = 15y + 12y \]
\[ 8x = 27y \]

5. **Express \(x\) in terms of \(y\)**:
\[ \frac{x}{y} = \frac{27}{8} \]

6. **Let \(x = 27k\) and \(y = 8k\)** for some constant \(k\).

7. **Substitute \(x\) and \(y\) into the expression**:
\[ \frac{\sqrt[3]{x} + \sqrt[3]{y}}{\sqrt[3]{x} - \sqrt[3]{y}} = \frac{\sqrt[3]{27k} + \sqrt[3]{8k}}{\sqrt[3]{27k} - \sqrt[3]{8k}} \]

8. **Calculate the cube roots**:
\[ \sqrt[3]{27k} = 3\sqrt[3]{k}, \quad \sqrt[3]{8k} = 2\sqrt[3]{k} \]

9. **Substitute these values back into the expression**:
\[ \frac{3\sqrt[3]{k} + 2\sqrt[3]{k}}{3\sqrt[3]{k} - 2\sqrt[3]{k}} = \frac{(3 + 2)\sqrt[3]{k}}{(3 - 2)\sqrt[3]{k}} = \frac{5\sqrt[3]{k}}{1\sqrt[3]{k}} = 5 \]

### Final Answer
The value of \(\frac{\sqrt[3]{x} + \sqrt[3]{y}}{\sqrt[3]{x} - \sqrt[3]{y}}\) is **5**.