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Multiplying Complex Numbers: (3 - 2i)(5 + 4i)
This article will guide you through the process of multiplying the complex numbers (3 - 2i) and (5 + 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).
Multiplication Process
To multiply complex numbers, we use the distributive property, similar to multiplying binomials.
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Expand the expression: (3 - 2i)(5 + 4i) = 3(5 + 4i) - 2i(5 + 4i)
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Distribute: = 15 + 12i - 10i - 8i²
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Substitute i² = -1: = 15 + 12i - 10i + 8
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Combine like terms: = (15 + 8) + (12 - 10)i
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Simplify: = 23 + 2i
Conclusion
Therefore, the product of (3 - 2i) and (5 + 4i) is 23 + 2i.
This process demonstrates how to multiply complex numbers by treating them like binomials, distributing, and simplifying using the definition of the imaginary unit.