(2-7i%5E4).jpeg "(3+5i)(2-7i^4)")
Multiplying Complex Numbers: (3 + 5i)(2 - 7i^4)
This article will guide you through the process of multiplying the complex numbers (3 + 5i) and (2 - 7i^4).
Understanding Complex Numbers
Complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit, defined as √-1.
Simplifying i^4
Before we begin multiplication, we need to simplify i^4. We know that:
- i^1 = i
- i^2 = -1
- i^3 = i^2 * i = -1 * i = -i
- i^4 = i^2 * i^2 = -1 * -1 = 1
Therefore, i^4 = 1.
Multiplication Process
Now we can multiply the complex numbers:
(3 + 5i)(2 - 7i^4)
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Substitute i^4 with 1: (3 + 5i)(2 - 7(1))
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Simplify the expression: (3 + 5i)(2 - 7)
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Distribute: (3 * 2) + (3 * -7) + (5i * 2) + (5i * -7)
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Simplify: 6 - 21 + 10i - 35i
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Combine real and imaginary terms: (6 - 21) + (10 - 35)i
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Final result: -15 - 25i
Conclusion
Therefore, the product of (3 + 5i) and (2 - 7i^4) is -15 - 25i. This process demonstrates how to multiply complex numbers and simplify expressions involving powers of the imaginary unit, i.