Simplifying Complex Expressions: (32i)(5+4i)(34i)^2
This article will guide you through simplifying the complex expression: (32i)(5+4i)(34i)^2. We will use the properties of complex numbers and algebraic operations to reach a solution.
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 √1.
Simplifying the Expression
Let's break down the simplification step by step:

Expand the first product: (32i)(5+4i) = 15 + 12i  10i  8i² Remember that i² = 1. Substitute this: = 15 + 12i  10i + 8 = 23 + 2i

Expand the second product: (34i)² = (34i)(34i) = 9  12i  12i + 16i² Substitute i² = 1: = 9  12i  12i  16 = 7  24i

Combine the results: (32i)(5+4i)(34i)² = (23 + 2i)  (7  24i)

Simplify: = 23 + 2i + 7 + 24i = 30 + 26i
Final Solution
Therefore, the simplified form of the expression (32i)(5+4i)(34i)^2 is 30 + 26i.