The expression provided is:

$\sqrt{2 + \sqrt{-2 + 2\sqrt{5}}} - \sqrt{2 - \sqrt{-2 + 2\sqrt{5}}}$

To simplify this expression, we'll approach it step by step.

Step 1: Simplify the Inner Expressions

We first focus on the expression inside the square roots:

$\sqrt{-2 + 2\sqrt{5}}$

This expression is complex because it contains a negative term under the square root. Let's denote this expression as:

$y = \sqrt{-2 + 2\sqrt{5}}$

The term inside the square root, $-2 + 2\sqrt{5}$, can be rewritten, but it cannot be simplified further into real numbers because it remains under the square root with a negative term. This makes $y$ an imaginary number.

Step 2: Simplify the Entire Expression

Given that the expression inside the square roots is imaginary, the overall expression becomes more challenging to simplify in terms of real numbers. However, let's denote:

$x = \sqrt{2 + \sqrt{-2 + 2\sqrt{5}}} - \sqrt{2 - \sqrt{-2 + 2\sqrt{5}}}$

Step 3: Square the Expression

To see if any simplifications occur when we square the expression $x$, we proceed as follows:

$x^2 = \left(\sqrt{2 + \sqrt{-2 + 2\sqrt{5}}} - \sqrt{2 - \sqrt{-2 + 2\sqrt{5}}}\right)^2$

Using the identity $(a - b)^2 = a^2 - 2ab + b^2$, we have:

$x^2 = \left(2 + \sqrt{-2 + 2\sqrt{5}}\right) + \left(2 - \sqrt{-2 + 2\sqrt{5}}\right) - 2\sqrt{\left(2 + \sqrt{-2 + 2\sqrt{5}}\right)\left(2 - \sqrt{-2 + 2\sqrt{5}}\right)}$

Simplify the first two terms:

$x^2 = 4 - 2\sqrt{\left(2 + \sqrt{-2 + 2\sqrt{5}}\right)\left(2 - \sqrt{-2 + 2\sqrt{5}}\right)}$

Now, the product under the square root can be simplified using the difference of squares:

$\left(2 + \sqrt{-2 + 2\sqrt{5}}\right)\left(2 - \sqrt{-2 + 2\sqrt{5}}\right) = 4 - \left(-2 + 2\sqrt{5}\right) = 6 - 2\sqrt{5}$

This gives:

$x^2 = 4 - 2\sqrt{6 - 2\sqrt{5}}$

However, this expression is still quite complicated and generally does not simplify to a nice, neat real number because of the imaginary and irrational components involved.

Step 4: Evaluate Simplifications

Because the square root of a negative number is imaginary, and no further simplification leads to a clear real number, the expression $x$ likely remains a complex number. Simplifying it further would typically involve complex number techniques, but it will not simplify to a zero or another simple integer.

Given the structure, without more advanced operations or context, this expression does not simplify easily into a simpler form or integer.

Conclusion

The expression remains complex and involves imaginary numbers due to the square root of a negative number within the inner square roots. This complicates further simplification, and in standard form, the expression remains as:

$\boxed{\sqrt{2 + \sqrt{-2 + 2\sqrt{5}}} - \sqrt{2 - \sqrt{-2 + 2\sqrt{5}}}}$

Would you like to explore this further or have any questions?

Related Questions

1. What are some common techniques for simplifying expressions with imaginary numbers?
2. How do complex numbers affect the simplification of square roots?
3. Can complex expressions like this one ever simplify to a real number?
4. What are some applications of complex numbers in physics or engineering?
5. How can the difference of squares be applied to more complex algebraic expressions?

Tip

When dealing with expressions involving square roots of negative numbers, always consider whether the problem might be in the domain of complex numbers. Simplifying such expressions may require working with imaginary numbers and understanding their properties.