https://file.aiquickdraw.com/mathbot/file/1729289691427upp7crv3.jpg

The given expression is:

\[
\sqrt{\left(\sqrt{7} - 3\right)^2} + \sqrt{7}
\]

### Step-by-step solution:

1. **Simplify the squared term**:
   \[
   \left(\sqrt{7} - 3\right)^2
   \]
   When we square this, the square and the square root will cancel out:
   \[
   = (\sqrt{7} - 3)(\sqrt{7} - 3)
   = 7 - 6\sqrt{7} + 9
   = 16 - 6\sqrt{7}
   \]

2. **Simplify the square root**:
   \[
   \sqrt{(\sqrt{7} - 3)^2}
   \]
   Since we are taking the square root of a perfect square, we just get:
   \[
   = \lvert \sqrt{7} - 3 \rvert
   \]
   Since \(\sqrt{7} \approx 2.65\), the expression inside the absolute value is negative. Therefore:
   \[
   = 3 - \sqrt{7}
   \]

3. **Final expression**:
   Now, add the result from the previous step to \(\sqrt{7}\):
   \[
   (3 - \sqrt{7}) + \sqrt{7}
   = 3
   \]

Thus, the final simplified result is:

\[
\boxed{3}
\]

Let me know if you'd like more details or if you have any questions!

### Related questions:
1. How do absolute values affect square roots and squares?
2. What happens when you square a binomial involving a square root?
3. Can you explain how to work with radicals more efficiently?
4. Why do square and square root operations cancel each other out?
5. How do you approximate \(\sqrt{7}\) or other irrational numbers?

### Tip:
Always check whether you're working with absolute values when simplifying square roots, especially when variables or expressions could be negative.

Simplify the expression: sqrt((sqrt(7) - 3)^2) + sqrt(7)

Learn how to simplify the expression sqrt((sqrt(7) - 3)^2) + sqrt(7) with a detailed step-by-step solution. This problem covers key concepts such as radicals, absolute values, and properties of square roots. Suitable for students in Grades 9-12.

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