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The image asks to "Take the conjugate of the square root expression: \(\sqrt{\frac{1}{2+2i} - \sqrt{2}}\)."

To find the conjugate of a complex expression, we typically replace each occurrence of \(i\) with \(-i\). Let's go through this step by step:

### Step-by-Step Solution

1. **Original Expression**:
   \[
   \sqrt{\frac{1}{2 + 2i} - \sqrt{2}}
   \]

2. **Identify the Complex Parts**:
   - The complex part of the expression is \(2 + 2i\).

3. **Find the Conjugate of the Complex Number \(2 + 2i\)**:
   - The conjugate of \(2 + 2i\) is \(2 - 2i\).

4. **Rewrite the Expression with the Conjugate**:
   To find the conjugate of the whole expression, substitute \(i\) with \(-i\) in the denominator:
   \[
   \sqrt{\frac{1}{2 - 2i} - \sqrt{2}}
   \]

5. **Result**:
   The conjugate of the expression \(\sqrt{\frac{1}{2 + 2i} - \sqrt{2}}\) is \(\sqrt{\frac{1}{2 - 2i} - \sqrt{2}}\).

### Summary

The conjugate of \(\sqrt{\frac{1}{2 + 2i} - \sqrt{2}}\) is \(\sqrt{\frac{1}{2 - 2i} - \sqrt{2}}\).

Would you like to see further steps, such as simplifying this expression or understanding conjugates in more detail?

Here are some related questions you might find interesting:
1. How do you simplify the expression \(\frac{1}{2 + 2i}\)?
2. What are the properties of conjugates in complex numbers?
3. How do you rationalize a complex denominator?
4. What is the effect of taking the conjugate on the modulus of a complex number?
5. How can you express \(\sqrt{2}\) in terms of complex numbers?

**Tip:** When finding the conjugate of a square root expression involving complex numbers, always ensure you replace \(i\) with \(-i\) in every occurrence.

Take the conjugate of the square root expression: √(1 / (2 + 2i) - √2).

Learn how to find the conjugate of the square root expression √(1 / (2 + 2i) - √2). This detailed guide covers the concept of conjugates in complex numbers and the steps required to solve the problem, suitable for students in Grades 11-12.

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