Divide and express the result in standard form. - 6i/(3 + 2i)

Verified step by step guidance

1. Step 1: Recognize that the denominator contains a complex number (3 + 2i). To simplify the division, multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of (3 + 2i) is (3 - 2i).
2. Step 2: Write the expression as (6i * (3 - 2i)) / ((3 + 2i) * (3 - 2i)). This step ensures that the denominator becomes a real number.
3. Step 3: Expand the numerator using the distributive property: 6i * 3 = 18i and 6i * (-2i) = -12i². Remember that i² = -1, so replace -12i² with 12.
4. Step 4: Expand the denominator using the difference of squares formula: (3 + 2i)(3 - 2i) = 3² - (2i)². Simplify this to 9 - (-4), which equals 13.
5. Step 5: Combine the results from the numerator and denominator. The numerator becomes (18i + 12), and the denominator is 13. Write the final expression in standard form as (12/13) + (18/13)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 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 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.