(3+5i)(2-7i^4)

2 min read Jun 16, 2024
(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)

  1. Substitute i^4 with 1: (3 + 5i)(2 - 7(1))

  2. Simplify the expression: (3 + 5i)(2 - 7)

  3. Distribute: (3 * 2) + (3 * -7) + (5i * 2) + (5i * -7)

  4. Simplify: 6 - 21 + 10i - 35i

  5. Combine real and imaginary terms: (6 - 21) + (10 - 35)i

  6. 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.

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