2.jpeg "(8+3i)2")
Expanding (8 + 3i)<sup>2</sup>
This article will walk through the process of expanding and simplifying the expression (8 + 3i)<sup>2</sup>, where 'i' represents the imaginary unit (√-1).
Understanding the Basics
- Imaginary Unit: The imaginary unit 'i' is defined as the square root of -1. This allows us to work with the square roots of negative numbers.
- Complex Numbers: A complex number is a number of the form a + bi, where a and b are real numbers and 'i' is the imaginary unit.
Expanding the Expression
To expand (8 + 3i)<sup>2</sup>, we use the FOIL method:
- First: 8 * 8 = 64
- Outer: 8 * 3i = 24i
- Inner: 3i * 8 = 24i
- Last: 3i * 3i = 9i<sup>2</sup>
Combining the terms: 64 + 24i + 24i + 9i<sup>2</sup>
Simplifying the Expression
Remember that i<sup>2</sup> = -1. Substituting this into our expression:
64 + 24i + 24i + 9(-1)
Simplifying further:
64 + 48i - 9
Final Result
Combining real and imaginary terms:
(8 + 3i)<sup>2</sup> = 55 + 48i
Therefore, the simplified form of (8 + 3i)<sup>2</sup> is 55 + 48i.