The expression given in the image is:

$\left(x^2 \sqrt{2}\right)^4 \cdot \left(x^2 \sqrt{2}\right)^{-4}$

Let's simplify this step by step.

Step 1: Simplify each term individually

For the first term:

$\left(x^2 \sqrt{2}\right)^4$

Apply the power to both the base $x^2$ and the constant $\sqrt{2}$:

$(x^2)^4 \cdot (\sqrt{2})^4$

Simplify further:

$x^{8} \cdot (\sqrt{2})^4 = x^8 \cdot (2^2) = x^8 \cdot 4$

So, the first term simplifies to:

$4x^8$

For the second term:

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

Similarly, apply the power to both the base $x^2$ and the constant $\sqrt{2}$:

$(x^2)^{-4} \cdot (\sqrt{2})^{-4}$

Simplify further:

$x^{-8} \cdot (\sqrt{2})^{-4} = x^{-8} \cdot 2^{-2} = x^{-8} \cdot \frac{1}{4}$

So, the second term simplifies to:

$\frac{1}{4x^8}$

Step 2: Multiply the simplified terms

Now multiply the two simplified expressions:

$4x^8 \cdot \frac{1}{4x^8}$

The $4$ and $\frac{1}{4}$ cancel out, and the $x^8$ and $x^{-8}$ cancel out as well, leaving:

$1$

Final Answer

The expression simplifies to:

$1$

Would you like further details or have any questions? Here are some related questions:

1. How do you simplify expressions with exponents?
2. What is the power rule in exponents?
3. How do you handle square roots raised to powers?
4. What is the importance of exponent rules in algebra?
5. How do you simplify complex expressions involving both variables and constants?

Tip: When simplifying expressions with exponents, always apply the power to every factor within the parentheses.