-(7-5i)%2B2i(9%2B12i).jpeg "(3+4i)-(7-5i)+2i(9+12i)")
Simplifying Complex Numbers: A Step-by-Step Guide
This article will guide you through the process of simplifying the complex number expression: (3 + 4i) - (7 - 5i) + 2i(9 + 12i).
Understanding Complex Numbers
Before we dive into the simplification, let's understand what complex numbers are. Complex numbers are numbers that can be expressed in the form a + bi, where:
- a and b are real numbers
- i is the imaginary unit, defined as the square root of -1 (i² = -1).
Simplifying the Expression
Step 1: Distribute the 2i
Begin by distributing the 2i across the parentheses: (3 + 4i) - (7 - 5i) + 2i(9 + 12i) = (3 + 4i) - (7 - 5i) + 18i + 24i²
Step 2: Substitute i² with -1
Remember that i² = -1. Substitute this value into the expression: (3 + 4i) - (7 - 5i) + 18i + 24i² = (3 + 4i) - (7 - 5i) + 18i + 24(-1)
Step 3: Simplify by Combining Like Terms
Now, simplify the expression by combining real and imaginary terms: (3 + 4i) - (7 - 5i) + 18i - 24 = (3 - 7 - 24) + (4 + 5 + 18)i
Step 4: Calculate the Final Result
Finally, combine the real and imaginary components: (3 - 7 - 24) + (4 + 5 + 18)i = -28 + 27i
Conclusion
Therefore, the simplified form of the complex number expression (3 + 4i) - (7 - 5i) + 2i(9 + 12i) is -28 + 27i. This process highlights the key steps in simplifying complex number expressions involving multiplication and addition/subtraction.