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Chapter 7: Problem 66

Divide and simplify. Write each answer in the form \(a+b i\). $$\frac{26}{5+i}$$

Short Answer

Expert verified
5 - i

Step by step solution

01

- Identify the Conjugate

To simplify the expression \(\frac{26}{5+i}\), multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(5+i\) is \(5-i\).
02

- Multiply by the Conjugate

Multiply both the numerator and the denominator by \(5-i\): \(\frac{26}{5+i} \times \frac{5-i}{5-i}\). This keeps the value of the expression the same.
03

- Expand the Numerator

Expand the numerator: \(26 \times (5-i) = 130 - 26i\).
04

- Expand the Denominator

Expand the denominator using the difference of squares formula: \((5+i)(5-i) = 5^2 - i^2 = 25 - (-1) = 26\).
05

- Simplify the Fraction

Now, simplify the fraction: \(\frac{130 - 26i}{26} = \frac{130}{26} - \frac{26i}{26} = 5 - i\).

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Key Concepts

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

conjugate
When dividing complex numbers, using the conjugate is key. The conjugate of a complex number helps eliminate the imaginary part in the denominator. For a complex number of the form \(a + bi\), the conjugate is \(a - bi\). Imagine you have the expression \(\frac{26}{5+i}\). To simplify it, multiply both the numerator and the denominator by the conjugate of \(5+i\), which is \(5-i\). This moves us towards a fraction that's easier to simplify. It's like turning a complicated road into a straight path.
difference of squares
When you multiply a complex number by its conjugate, you use the difference of squares formula, which states that \(a^2 - b^2\). In our example, \(5+i\) multiplied by \(5-i\) follows this pattern. Here, it becomes \[ (5+i)(5-i) = 5^2 - (i^2) = 25 - (-1) = 25 + 1 = 26. \] The imaginary parts cancel out, and the denominator becomes a real number. This makes your fraction simpler and cleaner, resulting in \(5 - i\) after simplification.
simplifying complex fractions
Simplifying complex fractions involves breaking them down to simpler forms. Remember the example \(\frac{26}{5+i}\)? First, we multiplied both top and bottom by the conjugate, \(5-i\). Next, we expanded both the numerator (getting \(130 - 26i\)) and the denominator (getting \(26\)). Now, divide each term in the numerator by the denominator to get \[ \frac{130 - 26i}{26} = \frac{130}{26} - \frac{26i}{26} = 5 - i. \] This final result \(5 - i\) is much simpler and is in the desired format \(a + bi\).

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