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To solve the problem, we are given \( x = 3 + \sqrt{8} \) and we need to find the value of \( x^2 + \frac{1}{x^2} \).

### Step-by-Step Solution:

1. **Find \( x^2 \):**

   We have:
   \[
   x = 3 + \sqrt{8}
   \]

   Squaring both sides:
   \[
   x^2 = (3 + \sqrt{8})^2
   \]

   Expand the square:
   \[
   x^2 = 3^2 + 2 \cdot 3 \cdot \sqrt{8} + (\sqrt{8})^2
   \]

   Calculate each term:
   \[
   x^2 = 9 + 6\sqrt{8} + 8
   \]

   Combine like terms:
   \[
   x^2 = 17 + 6\sqrt{8}
   \]

2. **Find \( \frac{1}{x} \):**

   To find \( \frac{1}{x} \), notice that:
   \[
   x = 3 + \sqrt{8}
   \]

   Then, the conjugate is \( 3 - \sqrt{8} \). Multiply and divide by the conjugate:
   \[
   \frac{1}{x} = \frac{1}{3 + \sqrt{8}} \times \frac{3 - \sqrt{8}}{3 - \sqrt{8}}
   \]

   The denominator simplifies to:
   \[
   (3 + \sqrt{8})(3 - \sqrt{8}) = 3^2 - (\sqrt{8})^2 = 9 - 8 = 1
   \]

   So:
   \[
   \frac{1}{x} = 3 - \sqrt{8}
   \]

   Now, square \( \frac{1}{x} \):
   \[
   \left( \frac{1}{x} \right)^2 = (3 - \sqrt{8})^2
   \]

   Expand the square:
   \[
   \left( \frac{1}{x} \right)^2 = 3^2 - 2 \cdot 3 \cdot \sqrt{8} + (\sqrt{8})^2
   \]

   Calculate each term:
   \[
   \left( \frac{1}{x} \right)^2 = 9 - 6\sqrt{8} + 8
   \]

   Combine like terms:
   \[
   \frac{1}{x^2} = 17 - 6\sqrt{8}
   \]

3. **Find \( x^2 + \frac{1}{x^2} \):**

   Now, add \( x^2 \) and \( \frac{1}{x^2} \):
   \[
   x^2 + \frac{1}{x^2} = (17 + 6\sqrt{8}) + (17 - 6\sqrt{8})
   \]

   The terms with \( \sqrt{8} \) cancel out:
   \[
   x^2 + \frac{1}{x^2} = 17 + 17 = 34
   \]

### Final Answer:
\[
x^2 + \frac{1}{x^2} = 34
\]

Would you like more details on any step, or do you have any questions? 

### Further Related Questions:
1. How do you simplify expressions involving radicals like \( \sqrt{8} \)?
2. What is the conjugate of a binomial, and why is it used in simplifying fractions?
3. Can you explain the process of rationalizing the denominator?
4. How would this problem change if \( x = 3 - \sqrt{8} \)?
5. What are the properties of exponents that are helpful in problems like this?

### Tip:
When dealing with expressions involving square roots, always check if you can use conjugates to simplify calculations.

If x = 3 + √8, then find the value of x² + 1/x².

Learn how to find the value of x² + 1/x² when x = 3 + √8. This problem involves squaring binomials, simplifying radicals, and using conjugates to rationalize the denominator. Suitable for Grades 9-10 algebra students.

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