2select-the-product.jpeg "(8 – 5i)2select The Product")
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.