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The expression you have provided is:

\[
\frac{4}{\sqrt{7} - \sqrt{5}}
\]

To simplify this, we can rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(\sqrt{7} - \sqrt{5}\) is \(\sqrt{7} + \sqrt{5}\).

### Step-by-step process:

1. Multiply the numerator and the denominator by the conjugate \(\sqrt{7} + \sqrt{5}\):
\[
\frac{4}{\sqrt{7} - \sqrt{5}} \times \frac{\sqrt{7} + \sqrt{5}}{\sqrt{7} + \sqrt{5}} = \frac{4(\sqrt{7} + \sqrt{5})}{(\sqrt{7})^2 - (\sqrt{5})^2}
\]

2. Simplify the denominator using the difference of squares:
\[
(\sqrt{7})^2 - (\sqrt{5})^2 = 7 - 5 = 2
\]

3. Now the expression becomes:
\[
\frac{4(\sqrt{7} + \sqrt{5})}{2}
\]

4. Simplify by dividing both the numerator and denominator by 2:
\[
2(\sqrt{7} + \sqrt{5})
\]

Thus, the simplified form of the expression is:
\[
2(\sqrt{7} + \sqrt{5})
\]

Would you like more details or have any questions?

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Here are 5 related questions:
1. How do you rationalize a denominator with multiple terms?
2. What is the significance of the difference of squares in simplification?
3. How can this technique be applied to more complex expressions?
4. Can you explain why multiplying by the conjugate eliminates square roots in the denominator?
5. How would the approach differ if both terms in the denominator were negative?

**Tip:** Always look for conjugates when simplifying radicals in the denominator.

Simplify the expression \(\frac{4}{\sqrt{7} - \sqrt{5}}\).

Learn how to simplify the expression \(\frac{4}{\sqrt{7} - \sqrt{5}}\) by rationalizing the denominator using the difference of squares method. This step-by-step solution is suitable for students in Grades 9-11, covering important algebraic concepts and radical expressions.

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