(2i-i^2)^2+(1-3i)^3

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Expressions: (2i - i^2)^2 + (1 - 3i)^3

This article will guide you through the process of simplifying the complex expression (2i - i^2)^2 + (1 - 3i)^3. We'll break down the steps and explain the rules involved.

Understanding Complex Numbers

Complex numbers are numbers of the form a + bi, where:

• a is the real part.
• b is the imaginary part.
• i is the imaginary unit, defined as the square root of -1 (i² = -1).

Simplifying the Expression

1. Simplify inside the parentheses:

   • (2i - i^2)
     • Remember that i² = -1.
     • This becomes (2i + 1).
   • (1 - 3i)
     • This remains as it is.

2. Expand the powers:

   • (2i + 1)²
     • Use the FOIL method: (2i + 1)(2i + 1) = 4i² + 2i + 2i + 1
     • Substitute i² = -1: -4 + 4i + 1 = -3 + 4i
   • (1 - 3i)³
     • This can be expanded using the binomial theorem or by multiplying step-by-step.
     • (1 - 3i)(1 - 3i)(1 - 3i) = (1 - 6i + 9i²)(1 - 3i)
     • Substitute i² = -1: (-8 - 6i)(1 - 3i) = -8 + 24i - 6i + 18i²
     • Simplify: -26 + 18i

3. Combine the simplified terms:

   • (-3 + 4i) + (-26 + 18i)
     • Combine real parts and imaginary parts separately: (-3 - 26) + (4 + 18)i
     • Simplify: -29 + 22i

Conclusion

Therefore, the simplified form of the complex expression (2i - i^2)² + (1 - 3i)³ is -29 + 22i.