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(1/1-2i+3/1+i)(3+4i/2-4i)

3 min read Jun 16, 2024
(1/1-2i+3/1+i)(3+4i/2-4i)

Simplifying Complex Expressions: A Step-by-Step Guide

This article will guide you through simplifying the complex expression:

(1/1-2i+3/1+i)(3+4i/2-4i)

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.

Step 1: Simplify the Fractions

We start by simplifying the individual fractions in the expression. To do this, we multiply both the numerator and denominator of each fraction by the conjugate of its denominator:

  • 1/(1-2i): Multiply by (1+2i)/(1+2i): (1 * (1+2i)) / ((1-2i) * (1+2i)) = (1+2i) / (1+4) = (1+2i)/5

  • 3/(1+i): Multiply by (1-i)/(1-i): (3 * (1-i)) / ((1+i) * (1-i)) = (3-3i) / (1+1) = (3-3i)/2

Step 2: Combine the Simplified Fractions

Now we combine the simplified fractions:

(1/1-2i+3/1+i) = (1+2i)/5 + (3-3i)/2

To add fractions, we need a common denominator. The least common denominator for 5 and 2 is 10.

  • [(1+2i) * 2 + (3-3i) * 5] / 10
  • (2+4i + 15-15i) / 10
  • (17-11i) / 10

Step 3: Simplify the Second Complex Fraction

Now we simplify the second complex fraction: (3+4i)/(2-4i)

Multiply the numerator and denominator by the conjugate of the denominator (2+4i):

  • ((3+4i) * (2+4i)) / ((2-4i) * (2+4i))
  • (6 + 12i + 8i + 16i^2) / (4 + 16)
  • (6 + 20i - 16) / 20
  • (-10 + 20i) / 20
  • (-1/2 + i)

Step 4: Multiply the Simplified Complex Numbers

Finally, we multiply the two simplified complex numbers:

(17-11i)/10 * (-1/2 + i) = (-17/20 + 11i/20) + (-17i/10 - 11/10)

Step 5: Combine Real and Imaginary Parts

Combining the real and imaginary terms:

(-17/20 - 11/10) + (11i/20 - 17i/10) = (-39/20 - 23i/20)

Therefore, the simplified form of the expression (1/1-2i+3/1+i)(3+4i/2-4i) is (-39/20 - 23i/20).

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