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Multiplying Complex Numbers: (7 + 5i)(8 - 6i)
This article will guide you through multiplying the complex numbers (7 + 5i) and (8 - 6i).
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 of Complex Numbers
To multiply complex numbers, we treat them like binomials and use the distributive property (or the FOIL method).
Let's multiply (7 + 5i) and (8 - 6i):
(7 + 5i)(8 - 6i) = 7(8) + 7(-6i) + 5i(8) + 5i(-6i)
Simplify by performing the multiplication:
= 56 - 42i + 40i - 30i²
Remember that i² = -1. Substitute this value:
= 56 - 42i + 40i - 30(-1)
Combine real and imaginary terms:
= (56 + 30) + (-42 + 40)i
Finally, simplify:
= 86 - 2i
Conclusion
Therefore, the product of (7 + 5i) and (8 - 6i) is 86 - 2i.