## Expanding (8 – 5i)<sup>2</sup>

This problem involves squaring a complex number. Here's how to solve it:

**Understanding Complex Numbers**

Complex numbers are numbers that can be expressed in the form *a + bi*, where *a* and *b* are real numbers, and *i* is the imaginary unit, defined as the square root of -1 (i.e., i<sup>2</sup> = -1).

**Expanding the Expression**

We can expand (8 – 5i)<sup>2</sup> using the FOIL method (First, Outer, Inner, Last):

(8 – 5i)<sup>2</sup> = (8 – 5i)(8 – 5i)

= 8 * 8 + 8 * (-5i) + (-5i) * 8 + (-5i) * (-5i)

= 64 – 40i – 40i + 25i<sup>2</sup>

**Simplifying the Expression**

Since i<sup>2</sup> = -1, we can substitute:

= 64 – 40i – 40i + 25(-1)

= 64 – 40i – 40i – 25

= 39 – 80i

**Therefore, the product of (8 – 5i)<sup>2</sup> is 39 – 80i.**