Let's break down the expression inside the square root:

We are asked to find the square root of:

$\sqrt{z^2 \cdot (1 + z^2) + (1 + z^2)}$

Step 1: Factor out the common term $(1 + z^2)$

We notice that both terms inside the square root have the factor $(1 + z^2)$. So, we can factor it out:

$\sqrt{(1 + z^2) \cdot (z^2 + 1)}$

Step 2: Simplify the expression

The expression now becomes:

$\sqrt{(1 + z^2)^2}$

Step 3: Take the square root

Taking the square root of $(1 + z^2)^2$ simply gives us:

$1 + z^2$

Thus, the square root of $z^2 \cdot (1 + z^2) + (1 + z^2)$ simplifies to:

$1 + z^2$

Final Answer:

$\boxed{1 + z^2}$

Would you like a more detailed breakdown of any steps? Let me know if you have any other questions!

Related Questions:

1. How would the solution change if the expression had a negative sign instead of a plus sign inside the square root?
2. Can you find the square root of other types of binomial expressions?
3. How does factoring help in simplifying square roots?
4. What happens when you expand $(1 + z^2)^2$?
5. Can you find the square root of expressions involving more than two terms?

Tip: When simplifying square roots, look for common factors or perfect squares to make the process easier.