If ` x ^(2) = 11 + 2 sqrt(30 ) , ` find :
` (1)/(x)`

Answer

To solve the problem, we need to find the value of \( \frac{1}{x} \) given that \( x^2 = 11 + 2\sqrt{30} \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ x^2 = 11 + 2\sqrt{30} \]

2. **Express \( x \) in terms of the square root:**
\[ x = \sqrt{11 + 2\sqrt{30}} \]

3. **Rewrite \( 11 \) as \( 6 + 5 \) to facilitate the use of the perfect square formula:**
\[ x = \sqrt{6 + 5 + 2\sqrt{30}} \]

4. **Recognize that \( \sqrt{30} = \sqrt{6} \cdot \sqrt{5} \) and use the identity \( (a + b)^2 = a^2 + b^2 + 2ab \):**
\[ x = \sqrt{(\sqrt{6} + \sqrt{5})^2} \]

5. **Simplify the expression:**
\[ x = \sqrt{6} + \sqrt{5} \]

6. **Now, find \( \frac{1}{x} \):**
\[ \frac{1}{x} = \frac{1}{\sqrt{6} + \sqrt{5}} \]

7. **Rationalize the denominator:**
\[ \frac{1}{x} = \frac{1}{\sqrt{6} + \sqrt{5}} \cdot \frac{\sqrt{6} - \sqrt{5}}{\sqrt{6} - \sqrt{5}} = \frac{\sqrt{6} - \sqrt{5}}{(\sqrt{6})^2 - (\sqrt{5})^2} \]

8. **Calculate the denominator:**
\[ (\sqrt{6})^2 - (\sqrt{5})^2 = 6 - 5 = 1 \]

9. **Final expression for \( \frac{1}{x} \):**
\[ \frac{1}{x} = \sqrt{6} - \sqrt{5} \]

### Final Answer:
\[ \frac{1}{x} = \sqrt{6} - \sqrt{5} \]