Find each quotient. Write answers in standard form. 2 / 3i

Verified step by step guidance

1. Identify the problem: We need to divide \( \frac{2}{3i} \) and express the result in standard form \( a + bi \).
2. To eliminate the imaginary unit \( i \) from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 3i \) is \( -3i \).
3. Multiply the numerator and the denominator by \( -3i \): \( \frac{2}{3i} \times \frac{-3i}{-3i} \).
4. Simplify the expression: The numerator becomes \( 2 \times (-3i) = -6i \) and the denominator becomes \( 3i \times (-3i) = -9i^2 \).
5. Recall that \( i^2 = -1 \), so substitute \( -1 \) for \( i^2 \) in the denominator, resulting in \( -9(-1) = 9 \). Now, simplify the expression to get the quotient in standard form.

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 involving imaginary units, such as division.

Dividing Complex Numbers

To divide complex numbers, one typically multiplies 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 unit from the denominator, allowing the result to be expressed in standard form.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where 'a' and 'b' are real numbers. When performing operations with complex numbers, it is important to express the final answer in this form to clearly identify the real and imaginary components, facilitating further calculations or interpretations.