Simplifying Complex Expressions: (6+2i)(62i)(34i)(3+4i)
This article will demonstrate how to simplify the complex expression (6+2i)(62i)(34i)(3+4i).
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 (i² = 1).
Simplifying the Expression
To simplify the given expression, we will use the following properties:
 Difference of Squares: (a + b)(a  b) = a²  b²
 i² = 1
Let's break down the expression stepbystep:

(6+2i)(62i): Applying the difference of squares property, we get: (6+2i)(62i) = 6²  (2i)² = 36  4i²

(34i)(3+4i): Similarly, applying the difference of squares property: (34i)(3+4i) = 3²  (4i)² = 9  16i²

Substituting i² = 1:
 36  4i² = 36  4(1) = 36 + 4 = 40
 9  16i² = 9  16(1) = 9 + 16 = 25

Combining the results: (6+2i)(62i)(34i)(3+4i) = 40  25 = 15
Conclusion
Therefore, the simplified form of the complex expression (6+2i)(62i)(34i)(3+4i) is 15. This example showcases how utilizing the properties of complex numbers and algebraic manipulations can lead to a simplified result.