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.