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!

Learn how to simplify the expression ((2 - √5)^2)^1/2 step by step. This solution covers binomial expansion, simplifying square roots, and handling irrational numbers. Suitable for high school students (Grades 9-12).

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