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

### Step-by-step solution:

1. **Write the expression:**
   \[
   \frac{2\sqrt{7}}{3 + \sqrt{7}}
   \]

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

3. **Simplify the denominator:**
   \[
   (3 + \sqrt{7})(3 - \sqrt{7}) = 3^2 - (\sqrt{7})^2 = 9 - 7 = 2
   \]
   So, the denominator becomes 2.

4. **Expand the numerator:**
   \[
   2\sqrt{7}(3 - \sqrt{7}) = 2\sqrt{7} \times 3 - 2\sqrt{7} \times \sqrt{7}
   \]
   \[
   = 6\sqrt{7} - 2 \times 7 = 6\sqrt{7} - 14
   \]

5. **Rewrite the expression:**
   \[
   \frac{6\sqrt{7} - 14}{2}
   \]

6. **Simplify the final result:**
   \[
   \frac{6\sqrt{7}}{2} - \frac{14}{2} = 3\sqrt{7} - 7
   \]

### Final Answer:
\[
3\sqrt{7} - 7
\]

Would you like further clarification on any steps?

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Here are 5 related questions for further practice:

1. Rationalize \(\frac{4\sqrt{5}}{2 + \sqrt{5}}\).
2. Simplify \(\frac{\sqrt{3}}{1 + \sqrt{3}}\) by rationalizing the denominator.
3. Rationalize and simplify \(\frac{5}{4 + \sqrt{2}}\).
4. Simplify \(\frac{3\sqrt{2}}{\sqrt{2} + \sqrt{3}}\).
5. Solve \(\frac{7}{5 - \sqrt{3}}\) by rationalizing the denominator.

**Tip:** Always multiply by the conjugate to eliminate square roots from denominators.

Learn how to rationalize the denominator of the expression (2√7)/(3 + √7) by multiplying with the conjugate. This step-by-step solution involves the difference of squares theorem and simplifying radical expressions. Suitable for students in Grades 9-11.

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