To simplify the expression \( \frac{1}{\sqrt{5+\sqrt{68}}} \), follow these steps:

1. **Simplify \( \sqrt{68} \):**
   \[
   \sqrt{68} = \sqrt{4 \times 17} = 2\sqrt{17}
   \]

2. **Substitute \( 2\sqrt{17} \) back into the original expression:**
   \[
   \frac{1}{\sqrt{5+\sqrt{68}}} = \frac{1}{\sqrt{5 + 2\sqrt{17}}}
   \]

3. **Approximate the value inside the square root:**
   - \( \sqrt{17} \approx 4.123 \)
   - Therefore, \( 2\sqrt{17} \approx 2 \times 4.123 = 8.246 \)
   - Now, calculate \( 5 + 8.246 \approx 13.246 \)

4. **Simplify the square root of the sum:**
   \[
   \sqrt{13.246} \approx 3.64
   \]

5. **Find the reciprocal:**
   \[
   \frac{1}{3.64} \approx 0.275
   \]

Thus, the approximate value of \( \frac{1}{\sqrt{5 + \sqrt{68}}} \) is \( 0.275 \).

Would you like more details or have any questions?

Here are 5 numbered questions you might ask next:
1. Can you show the steps for rationalizing the denominator of an expression like \( \frac{1}{\sqrt{a+b}} \)?
2. What are the properties of nested square roots in algebra?
3. How do you approximate square roots more accurately?
4. What is the significance of rationalizing the denominator in mathematical expressions?
5. Can you explain the process of simplifying expressions with nested radicals?

**Tip:** When dealing with nested square roots, it's often useful to break them down into smaller components to simplify the expression step by step.

Learn how to simplify the expression \( \frac{1}{\sqrt{5+\sqrt{68}}} \) with detailed steps and approximations. This problem involves square roots and nested radicals, suitable for advanced high school or college-level students.

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