1. **State the problem:** Solve the equation $$9^{4x^2 - 1} = 1$$ for $x$. 2. **Recall the rule:** Any number (except 0) raised to the power 0 equals 1, i.e., $$a^0 = 1$$ for $a \neq 0$. 3. Since the right side is 1, the exponent must be 0: $$4x^2 - 1 = 0$$ 4. Solve for $x$: $$4x^2 = 1$$ $$x^2 = \frac{1}{4}$$ 5. Take the square root of both sides: $$x = \pm \sqrt{\frac{1}{4}}$$ $$x = \pm \frac{1}{2}$$ **Final answer:** $$x = \frac{1}{2} \text{ or } x = -\frac{1}{2}$$