%5E2.jpeg "(8+3i)^2")
Expanding and Simplifying (8 + 3i)^2
This article will guide you through the process of expanding and simplifying the expression (8 + 3i)².
Understanding Complex Numbers
Before we begin, let's quickly review complex numbers. A complex number is a number 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.
Expanding the Expression
To expand (8 + 3i)², we can use the FOIL (First, Outer, Inner, Last) method or simply distribute:
(8 + 3i)² = (8 + 3i)(8 + 3i)
Using FOIL:
- First: 8 * 8 = 64
- Outer: 8 * 3i = 24i
- Inner: 3i * 8 = 24i
- Last: 3i * 3i = 9i²
Combining the terms:
64 + 24i + 24i + 9i²
Simplifying the Expression
Remember that i² = -1. Substitute this into our expression:
64 + 24i + 24i + 9(-1)
Simplify further:
64 + 48i - 9
Final Result:
(8 + 3i)² = 55 + 48i
Therefore, the simplified form of (8 + 3i)² is 55 + 48i.