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Chapter 1: Problem 26

Divide and express the result in standard form. $$ \frac{-6 i}{3+2 i} $$

Short Answer

Expert verified
The division of the given complex numbers results in \( \frac{12}{5} - \frac{18}{5}i \)

Step by step solution

01

Identify the Numerator and the Denominator

The numerator of the given complex fraction is \(-6 i\) and the denominator is \(3+2 i\). We aim to divide these two to get the result in standard form.
02

Multiply by the Conjugate

To eliminate the imaginary part from the denominator, multiply both numerator and denominator by the conjugate of the denominator, which is \(3 - 2i\). The fraction then becomes \[ \frac{-6i(3 - 2i)}{(3 + 2i)(3 - 2i)} \]
03

Simplify the Numerator

Multiplying the numbers in the numerator, we get: \[ -6i(3) + -6i(-2i) = -18i + 12 \], which simplifies to \(12 - 18i \)
04

Simplify the Denominator

Multiplying the numbers in the denominator, we get: \[ (3)(3) + (3)(-2i) + (2i)(3) + (2i)(-2i) = 9 - 6i + 6i - 4 = 5 \]
05

Divide by the Real Number

Now, simply divide the real and imaginary parts of the numerator by the real number in the denominator. The fraction becomes \( \frac{12}{5} - \frac{18i}{5}\)
06

Express in Standard Form

The standard form for complex numbers is \(a + bi\), so our final answer is \( \frac{12}{5} - \frac{18}{5}i \)

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