The image contains a mathematical expression. I'll extract and solve it step by step.

Extracted Expression

The problem seems to involve powers and square roots. Here's the expression as I interpret it:

$\left(\sqrt{2} - 1\right)^4 + \left(\sqrt{2} + 1\right)^4$

Solution:

Let $a = \sqrt{2}$. Then, the expression becomes:

$\left(a - 1\right)^4 + \left(a + 1\right)^4$

Step 1: Expand both terms using the binomial theorem

Using the binomial theorem, $(x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4$.

1. Expand $(a - 1)^4$: $(a - 1)^4 = a^4 - 4a^3 + 6a^2 - 4a + 1$

2. Expand $(a + 1)^4$: $(a + 1)^4 = a^4 + 4a^3 + 6a^2 + 4a + 1$

Step 2: Add the two expansions

$\left(a - 1\right)^4 + \left(a + 1\right)^4 = \left(a^4 - 4a^3 + 6a^2 - 4a + 1\right) + \left(a^4 + 4a^3 + 6a^2 + 4a + 1\right)$

Combine like terms: $2a^4 + 12a^2 + 2$

Step 3: Substitute $a = \sqrt{2}$

1. Compute $a^2 = (\sqrt{2})^2 = 2$.
2. Substitute $a^2 = 2$ into the expression: $2a^4 + 12a^2 + 2 = 2(2^2) + 12(2) + 2$

Step 4: Simplify

$2(4) + 24 + 2 = 8 + 24 + 2 = 34$

Final Answer:

The value of the given expression is:

$\boxed{34}$

Let me know if you'd like further clarification or details!

Related Questions:

1. What is the binomial theorem, and how is it applied in similar problems?
2. How do you simplify expressions involving powers of square roots?
3. What happens if the powers in the problem are odd instead of even?
4. Can you generalize this problem for $(\sqrt{n} - 1)^k + (\sqrt{n} + 1)^k$?
5. What is the relationship between symmetric expressions like this one?

Tip:

For symmetric expressions involving powers of $(a+b)$ and $(a-b)$, consider the structure of the expansion—the odd-power terms cancel, and the even-power terms double.