(2%2B3i%2F4%2B5i).jpeg "(2/1-i+3/1+i)(2+3i/4+5i)")
Simplifying Complex Number Expressions
This article will guide you through simplifying the complex number expression:
(2/1-i + 3/1+i)(2+3i/4+5i)
Let's break it down step by step:
1. Simplifying Individual Fractions
First, we need to simplify the fractions within the expression:
- 2/1-i: To get rid of the complex number in the denominator, we multiply both numerator and denominator by the conjugate of the denominator:
2/1-i * (1+i)/(1+i) = (2+2i)/(1-i^2) = (2+2i)/2 = 1+i
- 3/1+i: We follow the same process:
3/1+i * (1-i)/(1-i) = (3-3i)/(1-i^2) = (3-3i)/2 = 1.5-1.5i
- 2+3i/4+5i: Again, multiply numerator and denominator by the conjugate:
(2+3i)/(4+5i) * (4-5i)/(4-5i) = (8 - 10i + 12i + 15)/(16+25) = (23+2i)/41
2. Combining Simplified Fractions
Now, our expression becomes:
(1+i + 1.5-1.5i)(23+2i/41)
3. Performing Multiplication
Let's multiply the terms in the first bracket:
(2.5 - 0.5i)(23+2i/41)
Now, we distribute the terms:
(2.5 * 23 + 2.5 * 2i/41 - 0.5i * 23 - 0.5i * 2i/41)
Simplifying:
(57.5 + 5i/41 - 11.5i - i^2/41)
Since i^2 = -1, we substitute:
(57.5 + 5i/41 - 11.5i + 1/41)
4. Final Simplification
Combining real and imaginary terms:
(57.5 + 1/41) + (5/41 - 11.5)i
Finally, we arrive at the simplified form:
57.56 + (-11.44)i
Therefore, the simplified expression of (2/1-i+3/1+i)(2+3i/4+5i) is 57.56 - 11.44i.