To solve the given equation:

$\left( \sqrt{2 - \sqrt{3}} \right)^x + \left( \sqrt{2 + \sqrt{3}} \right)^x = 4$

we can proceed with the following steps:

1. Identify potential values for $x$: Since the sum equals 4, we can check for simple values of $x$ that might simplify the equation.

2. Check if $x = 2$ satisfies the equation: Calculate $\left( \sqrt{2 - \sqrt{3}} \right)^2$ and $\left( \sqrt{2 + \sqrt{3}} \right)^2$.

   $\left( \sqrt{2 - \sqrt{3}} \right)^2 = 2 - \sqrt{3}$ $\left( \sqrt{2 + \sqrt{3}} \right)^2 = 2 + \sqrt{3}$

   Adding these two expressions: $(2 - \sqrt{3}) + (2 + \sqrt{3}) = 4$

   Therefore, the equation is satisfied when $x = 2$.

Hence, the solution to the equation is:

$\boxed{x = 2}$

Would you like further details or have any other questions?

Here are 5 more questions you might want to ask:

1. Can you solve a different Olympiad-level equation for me?
2. How do you approach solving equations with multiple variables?
3. What are some strategies for solving polynomial equations?
4. How do you handle solving inequalities involving square roots?
5. Can you explain the properties of exponents and logarithms?

Tip: When dealing with equations involving square roots and exponents, always consider squaring both sides or substituting potential simple values to check for solutions efficiently.