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

Verified step by step guidance

1. Identify the problem: You need to simplify the quotient \(\frac{2}{3i}\) and write the answer in standard form, which means in the form \(a + bi\) where \(a\) and \(b\) are real numbers.
2. To eliminate the imaginary unit \(i\) from the denominator, multiply both the numerator and the denominator by the complex conjugate of the denominator. Since the denominator is \$3i\(, its conjugate is \)-3i$.
3. Multiply numerator and denominator by \(-3i\): \(\frac{2}{3i} \times \frac{-3i}{-3i} = \frac{2 \times (-3i)}{3i \times (-3i)}\).
4. Simplify the numerator: \(2 \times (-3i) = -6i\). Simplify the denominator: \(3i \times (-3i) = -9i^2\). Recall that \(i^2 = -1\), so replace \(i^2\) with \(-1\).
5. Simplify the denominator further: \(-9i^2 = -9 \times (-1) = 9\). So the expression becomes \(\frac{-6i}{9}\). Finally, write this as \(0 - \frac{6}{9}i\) and simplify the fraction if possible to get the standard form \(a + bi\).

Key Concepts

Complex Numbers and Standard Form

Complex numbers are expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit with i² = -1. Writing answers in standard form means expressing the result as a sum of a real part and an imaginary part.

Division of Complex Numbers

Dividing complex numbers involves multiplying the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part from the denominator. This process simplifies the expression into standard form.

Complex Conjugate

The complex conjugate of a complex number a + bi is a - bi. Multiplying by the conjugate removes the imaginary unit from the denominator, making it easier to simplify and write the quotient in standard form.