The expression provided is:

$\sqrt{z^2 + 20z + 100}$

Step 1: Simplify the expression inside the square root.

The quadratic expression $z^2 + 20z + 100$ is a perfect square trinomial. It can be factored as:

$z^2 + 20z + 100 = (z + 10)^2$

Step 2: Substitute this factorization into the square root.

$\sqrt{z^2 + 20z + 100} = \sqrt{(z + 10)^2}$

Step 3: Simplify the square root.

Since the square root of $(z + 10)^2$ is $z + 10$, we get:

$\sqrt{z^2 + 20z + 100} = z + 10$

Conclusion:

The simplified form of the expression is $z + 10$. The root of the expression is found by setting $z + 10 = 0$:

$z + 10 = 0 \quad \Rightarrow \quad z = -10$

Thus, the root is $z = -10$.

Would you like further details or have any additional questions? Here are five related questions to consider:

1. How do we identify perfect square trinomials?
2. What if the quadratic expression wasn't a perfect square—how would we handle the square root?
3. Can square roots always be simplified like this? When not?
4. How can you solve for roots of more complex quadratic equations?
5. What are the implications of complex roots when dealing with square roots?

Tip: When simplifying square roots, always check if the expression inside is a perfect square—it can save time and effort!