To find \((x^2 - \frac{1}{x^2})\) when \(x = 3 + \sqrt{2}\), let's go through the steps.

### Step 1: Calculate \(x^2\)
Given \(x = 3 + \sqrt{2}\), we square both sides:

\[
x^2 = (3 + \sqrt{2})^2 = 3^2 + 2 \cdot 3 \cdot \sqrt{2} + (\sqrt{2})^2
\]

\[
x^2 = 9 + 6\sqrt{2} + 2 = 11 + 6\sqrt{2}
\]

### Step 2: Calculate \(\frac{1}{x^2}\)
Now, let's find \(\frac{1}{x^2}\). First, recall that the conjugate of \(x = 3 + \sqrt{2}\) is \(3 - \sqrt{2}\), so:

\[
\frac{1}{x} = \frac{1}{3 + \sqrt{2}} \cdot \frac{3 - \sqrt{2}}{3 - \sqrt{2}} = \frac{3 - \sqrt{2}}{(3 + \sqrt{2})(3 - \sqrt{2})} = \frac{3 - \sqrt{2}}{9 - 2} = \frac{3 - \sqrt{2}}{7}
\]

Thus,

\[
\frac{1}{x} = \frac{3 - \sqrt{2}}{7}
\]

To square this:

\[
\frac{1}{x^2} = \left(\frac{3 - \sqrt{2}}{7}\right)^2 = \frac{(3 - \sqrt{2})^2}{7^2} = \frac{9 - 6\sqrt{2} + 2}{49} = \frac{11 - 6\sqrt{2}}{49}
\]

### Step 3: Calculate \(x^2 - \frac{1}{x^2}\)
Now subtract \(\frac{1}{x^2}\) from \(x^2\):

\[
x^2 - \frac{1}{x^2} = (11 + 6\sqrt{2}) - \frac{11 - 6\sqrt{2}}{49}
\]

We need to simplify this expression further. Let me handle the fraction carefully and compute the result.

The expression \(x^2 - \frac{1}{x^2}\) simplifies to:

\[
x^2 - \frac{1}{x^2} = \frac{528}{49} + \frac{300\sqrt{2}}{49}
\]

If you'd like, I can provide further details or verify any specific steps.

Would you like more details or have any questions? Here are five related questions you might find helpful:

1. How do you calculate the conjugate of a binomial involving a square root?
2. Can you explain the process of rationalizing a denominator?
3. What is the significance of conjugates in simplifying expressions?
4. How would you apply this method to a different binomial like \(5 + \sqrt{3}\)?
5. How can you check if your result is correct after performing algebraic operations?

**Tip:** When simplifying algebraic expressions, especially those involving square roots, using conjugates is a useful tool to eliminate radicals from denominators.

If (x = 3 + √2) then find (x² - 1/x²)

Discover how to find (x² - 1/x²) when x = 3 + √2. This problem involves conjugates, rationalizing binomials, and calculating square roots. Ideal for students in Grades 10-12, this step-by-step solution highlights key algebraic techniques.

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