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The image you uploaded contains a mathematical assignment focused on finding the values of expressions involving square roots and fractions.

Here’s a breakdown of the key parts of the assignment shown in the image:

1. **First Problem:** 
   - The task is to find the values of \( \alpha^2 \) and \( \beta^2 \).
   - \( \alpha \) and \( \beta \) are defined as:
     \[
     \alpha = \sqrt{5 + 2\sqrt{13}} - \sqrt{5 - 2\sqrt{13}}
     \]
     \[
     \beta = \sqrt{5 + 2\sqrt{13}} + \sqrt{5 - 2\sqrt{13}}
     \]
   - The solution involves squaring both expressions for \( \alpha \) and \( \beta \) to find \( \alpha^2 \) and \( \beta^2 \) respectively.

2. **Second Problem:**
   - You are given the task to prove that:
     \[
     \sqrt{12 + 7\sqrt{5}} = \sqrt{a} + \sqrt{b}
     \]
   - Here, \( a \) and \( b \) are positive integers.
   - You need to equate and solve to find the values of \( a \) and \( b \).

Would you like me to go through the steps to solve these problems in detail, or is there a particular part you'd like me to focus on?

Find the values of \( \alpha^2 \) and \( \beta^2 \) given \( \alpha = \sqrt{5 + 2\sqrt{13}} - \sqrt{5 - 2\sqrt{13}} \) and \( \beta = \sqrt{5 + 2\sqrt{13}} + \sqrt{5 - 2\sqrt{13}} \). Prove that \( \sqrt{12 + 7\sqrt{5}} = \sqrt{a} + \sqrt{b} \) where \( a \) and \( b \) are positive integers.

This problem involves finding and simplifying the square of expressions that include radical terms and square roots. It also requires proving an equation by simplifying radicals. This problem is suitable for students in Grades 10-12.

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