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Multiplying Complex Numbers: (4 - 7i)(4 + 7i)
This article will guide you through the process of multiplying the complex numbers (4 - 7i) and (4 + 7i).
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 use the distributive property, just like we do with real numbers.
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Expand the expression: (4 - 7i)(4 + 7i) = 4(4 + 7i) - 7i(4 + 7i)
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Distribute: = 16 + 28i - 28i - 49i²
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Simplify using i² = -1: = 16 + 28i - 28i - 49(-1) = 16 + 49
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Combine real and imaginary terms: = 65
Conclusion
The product of (4 - 7i) and (4 + 7i) is 65. Notice that the result is a real number. This is because (4 - 7i) and (4 + 7i) are complex conjugates of each other. Complex conjugates always result in a real number when multiplied.