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To rationalize the denominator of the given expression \(\frac{2}{7 - \sqrt{3}}\), we multiply both the numerator and denominator by the conjugate of the denominator, which is \(7 + \sqrt{3}\).

### Step-by-step solution:

1. Multiply the expression by \(\frac{7 + \sqrt{3}}{7 + \sqrt{3}}\):
   \[
   \frac{2}{7 - \sqrt{3}} \times \frac{7 + \sqrt{3}}{7 + \sqrt{3}} = \frac{2(7 + \sqrt{3})}{(7 - \sqrt{3})(7 + \sqrt{3})}
   \]

2. Apply the difference of squares formula to the denominator: 
   \[
   (7 - \sqrt{3})(7 + \sqrt{3}) = 7^2 - (\sqrt{3})^2 = 49 - 3 = 46
   \]

3. Now simplify the numerator:
   \[
   2(7 + \sqrt{3}) = 2 \times 7 + 2 \times \sqrt{3} = 14 + 2\sqrt{3}
   \]

4. The final expression is:
   \[
   \frac{14 + 2\sqrt{3}}{46}
   \]

5. Simplify by dividing both terms in the numerator by 2:
   \[
   \frac{14}{46} + \frac{2\sqrt{3}}{46} = \frac{7}{23} + \frac{\sqrt{3}}{23}
   \]

### Final Answer:
\[
\frac{7 + \sqrt{3}}{23}
\]

Would you like further details or have any questions?

Here are 5 related questions for deeper understanding:
1. How does multiplying by the conjugate help in rationalizing the denominator?
2. What is the difference of squares formula, and when is it used?
3. Can you rationalize the denominator of \(\frac{5}{2 + \sqrt{5}}\)?
4. How do you simplify expressions with surds (square roots)?
5. What are some common mistakes to avoid when rationalizing a denominator?

**Tip:** When rationalizing a denominator, always check for opportunities to simplify using the difference of squares formula when dealing with terms like \(a - \sqrt{b}\).

Rationalise the denominator of 2 / (7 - √3). Give your answer in its simplest form.

Learn how to rationalise the denominator of 2 / (7 - √3) using the conjugate method. Detailed step-by-step solution applying the difference of squares formula. Suitable for high school students in Grades 10-12.

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