To solve the equation $e^{x^2} = 1$, we need to find the values of $x$ that satisfy this equation.

The exponential function $e^y$ equals 1 when $y = 0$, because $e^0 = 1$.

Given $e^{x^2} = 1$, we can write:

$x^2 = 0$

Taking the square root of both sides, we get:

$x = 0$

Therefore, the solution to the equation $e^{x^2} = 1$ is:

$x = 0$

Would you like more details or have any questions?

Here are some related questions:

1. What is the value of $x$ in $e^x = 1$?
2. How do you solve $e^{x^2} = k$ for a general constant $k$?
3. What are the properties of the exponential function $e^x$?
4. Can $e^{x^2} = 1$ have any complex solutions?
5. What are the solutions to $e^{x^2} = -1$?
6. How would you graph the function $e^{x^2}$?
7. What happens to $e^{x^2}$ as $x$ approaches infinity?
8. How do you solve $e^{x^2} = e$ where $e$ is the base of the natural logarithm?

Tip: Remember that the exponential function $e^x$ is always positive and never zero.