2. Simplify the expression $x^{(x+1)^{2}} \cdot\left(x^{x-1}\right)^{x+1}: \frac{x^{x^{2}}}{x^{-2 x}}$ and then calculate the value of the expression for $x=\sqrt{2}$.
2. In the expression $x^{(x+1)^{2}}=x^{x^{2}+2 x+1}$, we use the squaring of a binomial in the exponent. In the power $\left(x^{x-1}\right)^{x+1}$, we apply the rule for raising a power to a power, thus obtaining $\left(x^{x-1}\right)^{x+1}=x^{x^{2}-1}$. We eliminate the fraction by considering the rule for dividing powers. We apply the rules for calculating with powers and simplify the expression to $x^{x^{2}}$. We substitute $\sqrt{2}$ and calculate the value of the expression.

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