(2+i)^3/2+3i

Jasonbradley

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Jun 16, 2024

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Calculating (2+i)^3 / (2+3i)

This problem involves complex number operations, specifically raising a complex number to a power and then dividing by another complex number. Let's break down the steps to solve it.

1. Calculating (2+i)^3

To find (2+i)^3, we can expand it using the binomial theorem or simply multiply it out:

• Using the binomial theorem: (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

  • In this case, a = 2 and b = i.
  • Substituting: (2+i)^3 = 2^3 + 3(2^2)(i) + 3(2)(i^2) + i^3
  • Simplifying: 8 + 12i - 6 - i = 2 + 11i

• Direct multiplication:

  • (2+i)^3 = (2+i)(2+i)(2+i)
  • Expanding: (2+i)(2+i) = 4 + 2i + 2i + i^2 = 3 + 4i
  • Multiplying again: (3+4i)(2+i) = 6 + 3i + 8i + 4i^2 = 2 + 11i

2. Dividing (2+11i) by (2+3i)

To divide complex numbers, we multiply both the numerator and denominator by the complex conjugate of the denominator:

• The complex conjugate of (2+3i) is (2-3i).

• Multiplying:

  • [(2+11i)(2-3i)] / [(2+3i)(2-3i)]
  • Expanding: [(4 + 22i - 33i^2) / (4 - 9i^2)]
  • Simplifying: (37 + 22i) / 13
  • Separating into real and imaginary parts: (37/13) + (22/13)i

Therefore, (2+i)^3 / (2+3i) = (37/13) + (22/13)i