In Exercises 21–28, divide and express the result in standard form. 2/(3 - i)

Verified step by step guidance

1. <Step 1: Identify the problem.> We need to divide \( \frac{2}{3 - i} \) and express the result in standard form, which is \( a + bi \) where \( a \) and \( b \) are real numbers.
2. <Step 2: Multiply by the conjugate.> To eliminate the imaginary part in the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 3 - i \) is \( 3 + i \). So, multiply \( \frac{2}{3 - i} \) by \( \frac{3 + i}{3 + i} \).
3. <Step 3: Simplify the numerator.> Distribute in the numerator: \( 2 \times (3 + i) = 6 + 2i \).
4. <Step 4: Simplify the denominator.> Use the difference of squares formula: \((3 - i)(3 + i) = 3^2 - i^2 = 9 - (-1) = 10\).
5. <Step 5: Write the result in standard form.> Combine the results from steps 3 and 4: \( \frac{6 + 2i}{10} = \frac{6}{10} + \frac{2i}{10} = \frac{3}{5} + \frac{1}{5}i \).

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit defined as the square root of -1. Understanding complex numbers is essential for performing operations such as addition, subtraction, multiplication, and division.

Division of Complex Numbers

Dividing complex numbers involves multiplying the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number a + bi is a - bi. This process eliminates the imaginary part in the denominator, allowing the result to be expressed in standard form, which is a + bi.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where a and b are real numbers. In this form, a represents the real part and b represents the imaginary part. Expressing complex numbers in standard form is crucial for clarity and consistency in mathematical communication, especially when performing further calculations or comparisons.