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Multiplying Complex Numbers: (9-2i)(3+i)
This article will guide you through the process of multiplying two complex numbers: (9-2i)(3+i).
Understanding Complex Numbers
Before we begin, let's quickly recap what complex numbers are. A complex number is a number 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.
The Multiplication Process
To multiply complex numbers, we use the distributive property (also known as FOIL - First, Outer, Inner, Last). This means we multiply each term in the first complex number by each term in the second complex number:
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(9-2i)(3+i) = (9 * 3) + (9 * i) + (-2i * 3) + (-2i * i)
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Simplify the terms: 27 + 9i - 6i - 2i²
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Remember that i² = -1: 27 + 9i - 6i + 2
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Combine real and imaginary terms: (27 + 2) + (9 - 6)i
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Final result: 29 + 3i
Conclusion
Therefore, the product of (9-2i) and (3+i) is 29 + 3i.
This example demonstrates how complex number multiplication is performed. The key is to remember the properties of the imaginary unit i, and use the distributive property to expand the product.