1. **State the problem:** Solve the equation $$p^4 - 625 = 0$$ for $p$. 2. **Rewrite the equation:** Recognize that $625 = 5^4$, so the equation becomes $$p^4 - 5^4 = 0$$. 3. **Use the difference of squares formula:** $$a^2 - b^2 = (a - b)(a + b)$$. Here, treat $$p^4 - 5^4$$ as $$(p^2)^2 - (5^2)^2$$, so $$p^4 - 625 = (p^2 - 25)(p^2 + 25) = 0$$. 4. **Solve each factor:** - For $$p^2 - 25 = 0$$, add 25 to both sides: $$p^2 = 25$$ Take the square root: $$p = \pm 5$$. - For $$p^2 + 25 = 0$$, subtract 25: $$p^2 = -25$$ Take the square root: $$p = \pm \sqrt{-25} = \pm 5i$$, where $i$ is the imaginary unit. 5. **Final answer:** The four solutions are $$p = 5, -5, 5i, -5i$$. These include two real roots and two purely imaginary roots.