Question: User: x = 1/x^2
1. **State the problem:** Solve the equation $$x = \frac{1}{x^2}$$ for $x$.
2. **Rewrite the equation:** Multiply both sides by $$x^2$$ to eliminate the denominator:
$$x \cdot x^2 = \frac{1}{x^2} \cdot x^2$$
which simplifies to:
$$x^3 = 1$$
3. **Solve for $x$:** To solve $$x^3 = 1$$, take the cube root of both sides:
$$x = \sqrt[3]{1}$$
4. **Evaluate the cube root:** The cube root of 1 is 1, so:
$$x = 1$$
5. **Check for extraneous solutions:** Substitute $x=1$ back into the original equation:
$$1 = \frac{1}{1^2} = 1$$
which is true.
6. **Consider domain restrictions:** Since the original equation has $x^2$ in the denominator, $x \neq 0$ to avoid division by zero.
7. **Final answer:** The solution is:
$$x = 1$$