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Chapter 3: Problem 77

Simplify. Write answers in the form \(a+b i,\) where \(a\) and \(b\) are real numbers. $$\frac{4-2 i}{1+i}+\frac{2-5 i}{1+i}$$

Short Answer

Expert verified
-\frac{1}{2} - \frac{13}{2}i

Step by step solution

01

Combine the fractions

Combine the two fractions by finding a common denominator. Since both fractions have the same denominator, simply add the numerators together: \[ \frac{4-2i}{1+i} + \frac{2-5i}{1+i} = \frac{(4-2i) + (2-5i)}{1+i} \ = \frac{(4+2) + (-2i - 5i)}{1+i} \ = \frac{6 - 7i}{1+i} \]
02

Multiply by the conjugate

Multiply the numerator and the denominator by the conjugate of the denominator to simplify: \[ \frac{6-7i}{1+i} \times \frac{1-i}{1-i} = \frac{(6-7i)(1-i)}{(1+i)(1-i)} \]
03

Simplify the numerator

Use the distributive property (FOIL method) to expand the numerator: \[ (6-7i)(1-i) = 6(1) + 6(-i) - 7i(1) - 7i(-i) \ = 6 - 6i - 7i + 7i^2 \ = 6 - 13i + 7(-1) \ = 6 - 13i - 7 \ = -1 - 13i \]
04

Simplify the denominator

The denominator simplifies as follows: \[ (1+i)(1-i) = 1 - i^2 \ = 1 - (-1) \ = 1 + 1 \ = 2 \]
05

Combine results

Simplify the division to get the final answer: \[ \frac{-1 - 13i}{2} \ = -\frac{1}{2} - \frac{13}{2}i \]So, the answer in the form of a + bi is: a = -\frac{1}{2}, b = -\frac{13}{2}

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Fraction
A complex fraction involves complex numbers in the numerator, the denominator, or both. Complex numbers are written in the form of \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit: \(i^2 = -1\).
To simplify a complex fraction, like \(\frac{4-2i}{1+i} + \frac{2-5i}{1+i}\), we start by combining the fractions together using common denominators. Since both fractions share the same denominator, we add the numerators and keep the denominator unchanged.
The combined fraction is: \(\frac{(4-2i) + (2-5i)}{1+i} = \frac{6-7i}{1+i}\). This paves the way to the next steps of simplification.
Conjugate
The conjugate of a complex number \(a + bi\) is \(a - bi\). It's essentially the complex number reflected over the real axis.
For example, the conjugate of \(1 + i\) is \(1 - i\). Conjugates are useful in simplifying complex fractions because they help eliminate the imaginary part in the denominator when multiplied together.
By multiplying both the numerator and denominator of \(\frac{6 - 7i}{1 + i}\) by the conjugate \(1 - i\), we simplify the complex fraction. This turns the denominator into a real number, facilitating further simplification.
FOIL Method
The FOIL method stands for First, Outer, Inner, Last. It's used to multiply two binomials. In the complex fraction \(\frac{6-7i}{1+i} \times \frac{1-i}{1-i}\), we focus on expanding the numerator:
Use FOIL Method:
- First (6 and 1): \(6 \cdot 1 = 6\)
- Outer (6 and -i): \(6 \cdot -i = -6i\)
- Inner (-7i and 1): \(-7i \cdot 1 = -7i\)
- Last (-7i and -i): \(-7i \cdot -i = 7i^2 = -7\)
Combine all terms: \(6 - 6i - 7i - 7 = -1 - 13i\). This simplifies the numerator into \(-1 - 13i\).
Simplify Complex Expression
After using the conjugate and the FOIL method, we simplify the denominator. The denominator \((1+i)(1-i)\) simplifies through multiplication:
\(1 \cdot 1 + 1 \cdot -i + i \cdot 1 + i \cdot -i = 1 - i^2 = 1 + 1 = 2\).
This gives us: \(\frac{-1 - 13i}{2}\).
Finally, divide each term by 2:
\(-\frac{1}{2} - \frac{13}{2}i\).
This results in the simplified expression in the form \(a + bi\), where \(a = -\frac{1}{2}\) and \(b = -\frac{13}{2}\).

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