To simplify the expression $((2 - 5^{1/2})^2)^{1/2}$, let's proceed step by step.

Step 1: Simplify the inner square

The expression inside the square root is $(2 - \sqrt{5})^2$.

Using the binomial expansion: $(a - b)^2 = a^2 - 2ab + b^2$ where $a = 2$ and $b = \sqrt{5}$.

$(2 - \sqrt{5})^2 = 2^2 - 2(2)(\sqrt{5}) + (\sqrt{5})^2$ $= 4 - 4\sqrt{5} + 5$ $= 9 - 4\sqrt{5}$

Step 2: Apply the square root

Now, we have $\sqrt{9 - 4\sqrt{5}}$. Since this is already in its simplest form, the expression cannot be simplified further in terms of real numbers.

Thus, the simplified result is: $\boxed{\sqrt{9 - 4\sqrt{5}}}$

Let me know if you'd like further details on this or have any questions!

Here are 5 related questions for deeper exploration:

1. What are some common methods for simplifying square root expressions?
2. How do you expand binomials with square roots?
3. What are the properties of irrational numbers like $\sqrt{5}$?
4. How does the binomial theorem help in expanding powers of expressions?
5. Can $\sqrt{9 - 4\sqrt{5}}$ be approximated numerically?

Tip: Always double-check whether an expression involving roots can be simplified further by factoring!