Problem 71 - Precalculus Student Solutions Manual 5th

Question

Find each quotient. Write the answer in standard form \(a+b i .\) $$\frac{2-i}{2+i}$$

Step-by-Step Solution

Answer

\( \frac{3}{5} - \frac{4}{5}i \)

Step 1: Identify the Conjugate of the Denominator

The given complex fraction is \( \frac{2-i}{2+i} \). To simplify this, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(2+i\) is \(2-i\).

Step 2: Multiply the Numerator and Denominator by the Conjugate

Multiply the numerator and the denominator by \(2-i\).\[\frac{(2-i)(2-i)}{(2+i)(2-i)}\]

Step 3: Expand the Numerator

Expand \((2-i)(2-i)\):\[(2-i)(2-i) = 2^2 - 2 \cdot i - 2 \cdot i + i^2 = 4 - 4i + i^2 \]Since \(i^2 = -1\), it simplifies to:\[4 - 4i - 1 = 3 - 4i\]

Step 4: Expand the Denominator

Expand \((2+i)(2-i)\):\[(2+i)(2-i) = 2^2 - (i)^2 = 4 - (-1) = 4 + 1 = 5\]

Step 5: Combine and Simplify

Now, we have:\[\frac{3-4i}{5}\]Separate the real and imaginary parts:\[\frac{3}{5} - \frac{4i}{5} = \frac{3}{5} - \frac{4}{5}i\]

Key Concepts

Conjugate

In complex numbers, the conjugate of a complex number is very important. The conjugate helps us simplify expressions and avoid imaginary numbers in the denominator.
To find the conjugate of a complex number, simply change the sign of the imaginary part.
For instance, the conjugate of a complex number \(a+bi\) is \(a-bi\).
In this exercise, the conjugate of \(2+i\) is \(2-i\).
By multiplying both the numerator and the denominator by the conjugate, we can remove the imaginary part from the denominator.

Standard Form

When working with complex numbers, the standard form is important for clarity. The standard form of a complex number is \(a+bi\) where \(a\) is the real part and \(b\) is the imaginary part.
After performing operations involving complex numbers, it's crucial to convert the result into standard form.
In our exercise, after simplifying \(\frac{2-i}{2+i}\), we ended up with \(\frac{3-4i}{5}\).
By splitting the fraction, we can clearly see the real and imaginary parts:
\(\frac{3}{5} - \frac{4}{5}i\).
This makes our final answer easy to understand and in the required standard form, \(a+bi\).

Simplifying Fractions

Simplifying fractions involving complex numbers might seem tricky at first, but it's straightforward with practice. To simplify fractions like \(\frac{2-i}{2+i}\), follow these steps:

• Find the conjugate of the denominator.
• Multiply the numerator and the denominator by this conjugate.
• Expand the resulting expressions.
• Simplify any squares of the imaginary unit \(i\).
• Combine and simplify the results.

For our exercise:
1. The conjugate of \(2+i\) is \(2-i\).
2. Multiply: \(\frac{(2-i)(2-i)}{(2+i)(2-i)}\).
3. Expand: Numerator: \((2-i)(2-i) = 4 - 4i + i^2 = 4 - 4i - 1 = 3-4i\), Denominator: \((2+i)(2-i) = 4 - i^2 = 4 + 1 = 5\).
4. Simplify: \(\frac{3-4i}{5}\).
Separate to real and imaginary parts: \(\frac{3}{5} - \frac{4}{5}i\).
This approach ensures we eliminate the imaginary part from the denominator and simplify the expression appropriately.