To simplify the expression \(\frac{2}{7 - 3\sqrt{2}}\), we need to rationalize the denominator. Here are the steps:

1. **Multiply the numerator and the denominator by the conjugate of the denominator**: The conjugate of \(7 - 3\sqrt{2}\) is \(7 + 3\sqrt{2}\).

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

2. **Simplify the denominator using the difference of squares**: 

\[ (7 - 3\sqrt{2})(7 + 3\sqrt{2}) = 7^2 - (3\sqrt{2})^2 = 49 - 18 = 31 \]

3. **Combine the results**:

\[ \frac{2(7 + 3\sqrt{2})}{31} = \frac{2 \cdot 7 + 2 \cdot 3\sqrt{2}}{31} = \frac{14 + 6\sqrt{2}}{31} \]

So, the simplified form of \(\frac{2}{7 - 3\sqrt{2}}\) is:

\[ \frac{14 + 6\sqrt{2}}{31} \]

Would you like a detailed explanation of any step? Here are 5 related questions you might find useful:

1. How do you rationalize the denominator of a fraction with a binomial containing a square root?
2. What is the purpose of multiplying by the conjugate in rationalization?
3. Can you provide more examples of rationalizing denominators with different conjugates?
4. How do you simplify expressions involving the difference of squares?
5. What are some common mistakes to avoid when rationalizing denominators?

**Tip:** Always check the simplified form of the expression to ensure that the numerator and the denominator are in their simplest forms and that no further simplification is possible.

Learn how to simplify the expression \(\frac{2}{7 - 3\sqrt{2}}\) with detailed steps. This problem involves rationalizing the denominator using the conjugate and the difference of squares. Suitable for high school students studying advanced algebra.

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