In Exercises 21–28, divide and express the result in standard form. - 6i/(3 + 2i)

Verified step by step guidance

1. Identify the problem as a division of complex numbers: \( \frac{6i}{3 + 2i} \).
2. To divide complex numbers, multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(3 + 2i\) is \(3 - 2i\).
3. Multiply both the numerator and the denominator by the conjugate: \( \frac{6i}{3 + 2i} \times \frac{3 - 2i}{3 - 2i} \).
4. Simplify the numerator: \(6i \times (3 - 2i)\). Use the distributive property to expand: \(6i \times 3 - 6i \times 2i\).
5. Simplify the denominator: \((3 + 2i)(3 - 2i)\). Use the difference of squares formula: \(a^2 - b^2\), where \(a = 3\) and \(b = 2i\).

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 and 'b' is the coefficient of the imaginary unit 'i', which is 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 from 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 expressed as a + bi, where 'a' and 'b' are real numbers. In this form, 'a' represents the real part and 'b' represents the imaginary part. Converting a complex number into standard form is crucial for clarity and further mathematical operations, making it easier to interpret and use in calculations.