Final answer:
The equation (p + 8)^2 = -64 leads to the solutions p = -8 + 8i and p = -8 - 8i using the square root property and recognizing the presence of an imaginary number.
Step-by-step explanation:
To solve for p using the square root property, we start with the equation (p + 8)^2 = -64. First, note that the square root of a negative number involves an imaginary number. Thus, taking the square root of both sides gives us:
√(p + 8)^2 = √-64
This simplifies to:
p + 8 = ±√-64
Since the square root of -64 can be written as √(64) √(-1) = 8i (where i is the imaginary unit), we have:
p + 8 = ± 8i
Subtracting 8 from both sides gives us:
p = -8 ± 8i
Therefore, the solutions for p are -8 + 8i and -8 - 8i.