-%E2%8B%85-(2-%E2%88%92-i).jpeg "(7 − 3i) ⋅ (2 − I)")
Multiplying Complex Numbers: (7 − 3i) ⋅ (2 − i)
This article will walk through the process of multiplying two complex numbers: (7 − 3i) ⋅ (2 − i).
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).
Multiplying Complex Numbers
Multiplying complex numbers is similar to multiplying binomials. We use the distributive property (or FOIL method) to expand the product.
Here's how to multiply (7 − 3i) ⋅ (2 − i):
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Distribute: (7 − 3i) ⋅ (2 − i) = 7(2 − i) − 3i(2 − i)
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Simplify: = 14 − 7i − 6i + 3i²
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Substitute i² = -1: = 14 − 7i − 6i + 3(-1)
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Combine real and imaginary terms: = (14 - 3) + (-7 - 6)i
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Final result: = 11 - 13i
Therefore, the product of (7 − 3i) ⋅ (2 − i) is 11 - 13i.
Conclusion
Multiplying complex numbers involves using the distributive property and substituting i² with -1. This results in a new complex number with both a real and imaginary component.