Let's solve the inequality \((\sqrt{3 + \sqrt{8}})^x + (\sqrt{3 - \sqrt{8}})^x \leq 34\).

### Step 1: Simplify the expressions
First, let's simplify the expressions inside the square roots.

- \(\sqrt{8} = 2\sqrt{2}\)
- So, \(\sqrt{3 + \sqrt{8}} = \sqrt{3 + 2\sqrt{2}}\)
- \(\sqrt{3 - \sqrt{8}} = \sqrt{3 - 2\sqrt{2}}\)

Now, let's denote:

\[
a = \sqrt{3 + 2\sqrt{2}} \quad \text{and} \quad b = \sqrt{3 - 2\sqrt{2}}
\]

The inequality becomes:

\[
a^x + b^x \leq 34
\]

### Step 2: Analyze the behavior of \(a\) and \(b\)
Since \(3 + 2\sqrt{2}\) is slightly greater than \(3\), \(a = \sqrt{3 + 2\sqrt{2}}\) is slightly greater than \(\sqrt{3}\).

Similarly, \(3 - 2\sqrt{2}\) is slightly less than \(3\), so \(b = \sqrt{3 - 2\sqrt{2}}\) is slightly less than \(\sqrt{3}\).

### Step 3: Evaluate specific values for \(x\)
Let's evaluate the expression for some specific values of \(x\).

1. **\(x = 0\):**
   \[
   a^0 + b^0 = 1 + 1 = 2
   \]
   This satisfies the inequality \(2 \leq 34\).

2. **\(x = 2\):**
   \[
   a^2 + b^2 = (3 + 2\sqrt{2}) + (3 - 2\sqrt{2}) = 6
   \]
   This satisfies the inequality \(6 \leq 34\).

3. **\(x = 4\):**
   \[
   a^4 + b^4 = (3 + 2\sqrt{2})^2 + (3 - 2\sqrt{2})^2
   \]
   Let's calculate:
   \[
   (3 + 2\sqrt{2})^2 = 9 + 12\sqrt{2} + 8 = 17 + 12\sqrt{2}
   \]
   \[
   (3 - 2\sqrt{2})^2 = 9 - 12\sqrt{2} + 8 = 17 - 12\sqrt{2}
   \]
   Adding these up:
   \[
   a^4 + b^4 = (17 + 12\sqrt{2}) + (17 - 12\sqrt{2}) = 34
   \]
   This satisfies the inequality \(34 \leq 34\).

### Step 4: Conclusion
The inequality holds for \(x = 4\), and because both functions \(a^x\) and \(b^x\) increase with increasing \(x\), \(x = 4\) is the maximum value for which the inequality holds.

Thus, the solution is:
\[
x \leq 4
\]

Do you want further details or have any questions?

### Related Questions:
1. What would happen if the inequality were \( \geq \) instead of \( \leq \)?
2. How does the behavior of \( \sqrt{3 + 2\sqrt{2}} \) compare to \( \sqrt{3 - 2\sqrt{2}} \)?
3. Can we solve this inequality using logarithms?
4. What if the inequality were an equation instead of an inequality?
5. How does the solution change if we substitute \(\sqrt{8}\) with a different value?

### Tip:
When solving inequalities involving roots and exponents, it's helpful to test specific values and consider the behavior of the function as \( x \) increases.

Learn how to solve the inequality (√(3+√8))^x + (√(3-√8))^x ≤ 34 with a detailed step-by-step solution. This problem involves exponents, inequalities, and square roots. Suitable for advanced high school students and beyond.

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