To simplify the expression \(\left( \frac{\sqrt{5} + \sqrt{2}}{2\sqrt{5} - \sqrt{2}} \right)^2\), we can proceed in steps as follows:

1. **Rewrite the expression**:
   \[
   \left( \frac{\sqrt{5} + \sqrt{2}}{2\sqrt{5} - \sqrt{2}} \right)^2
   \]

2. **Simplify the fraction inside the square** by rationalizing the denominator. To do this, multiply both the numerator and the denominator by the conjugate of the denominator, \(2\sqrt{5} + \sqrt{2}\):

   \[
   \frac{\sqrt{5} + \sqrt{2}}{2\sqrt{5} - \sqrt{2}} \cdot \frac{2\sqrt{5} + \sqrt{2}}{2\sqrt{5} + \sqrt{2}}
   \]

3. **Expand both the numerator and the denominator**:

   - In the **numerator**:
     \[
     (\sqrt{5} + \sqrt{2})(2\sqrt{5} + \sqrt{2}) = \sqrt{5} \cdot 2\sqrt{5} + \sqrt{5} \cdot \sqrt{2} + \sqrt{2} \cdot 2\sqrt{5} + \sqrt{2} \cdot \sqrt{2}
     \]
     Simplifying each term:
     \[
     = 2 \cdot 5 + \sqrt{10} + 2\sqrt{10} + 2 = 10

Explore the step-by-step solution to simplify the expression (√5 + √2 / 2√5 - √2)² by rationalizing the denominator and using algebraic techniques. This solution includes expanding and simplifying the terms and is suitable for students in Grades 10-12.

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