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:

   = 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$$

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.