(3-2i)(2+3i)/(1+2i)(2-i) In A+ib Form

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Numbers: (3-2i)(2+3i)/(1+2i)(2-i)

This article will guide you through the process of simplifying the complex number expression: (3-2i)(2+3i)/(1+2i)(2-i) and expressing it in the standard form a + bi.

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 (i.e., i² = -1).

Steps to Simplify

1. Multiply the Numerator: (3-2i)(2+3i) = (32) + (33i) + (-2i2) + (-2i3i) = 6 + 9i - 4i - 6i² Since i² = -1, we have: 6 + 9i - 4i + 6 = 12 + 5i

2. Multiply the Denominator: (1+2i)(2-i) = (12) + (1-i) + (2i2) + (2i-i) = 2 - i + 4i - 2i² Since i² = -1, we have: 2 - i + 4i + 2 = 4 + 3i

3. Divide the Numerator by the Denominator: (12 + 5i) / (4 + 3i)

4. Rationalize the Denominator: To eliminate the imaginary term in the denominator, we multiply both numerator and denominator by the conjugate of the denominator (4-3i):

   [(12 + 5i) * (4 - 3i)] / [(4 + 3i) * (4 - 3i)]

5. Expand and Simplify:

   • Numerator: (12 + 5i)(4 - 3i) = (124) + (12-3i) + (5i4) + (5i-3i) = 48 - 36i + 20i - 15i² = 48 - 16i + 15 = 63 - 16i
   • Denominator: (4 + 3i)(4 - 3i) = (44) + (4-3i) + (3i4) + (3i-3i) = 16 - 12i + 12i - 9i² = 16 + 9 = 25

6. Final Result: (63 - 16i) / 25 = (63/25) - (16/25)i

Therefore, the simplified form of the complex number expression (3-2i)(2+3i)/(1+2i)(2-i) in the standard a + bi form is (63/25) - (16/25)i.