QuestionEvaluate the quotient and express it as $a + b i$ : $\frac{3 - i}{6 + 4 i}$

Studdy Solution

STEP 1

Assumptions1. We are given a complex number in the form of a fraction, $\frac{3-i}{6+4i}$ . . We need to simplify this fraction and write it in the form of $a+bi$ , where $a$ and $b$ are real numbers, and $i$ is the imaginary unit.

STEP 2

To simplify a complex fraction, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number $a+bi$ is $a-bi$ . So, the conjugate of $6+4i$ is $6-4i$ .
$\frac{-i}{6+4i} \times \frac{6-4i}{6-4i}$

STEP 3

Now, multiply the numerators together and the denominators together. Remember that $i^2 = -1$ .
$\frac{(3-i)(6-i)}{(6+i)(6-i)}$

STEP 4

Expand the numerator and the denominator.
$\frac{18-12i-6i+4}{36-24i+24i-16}$

STEP 5

implify the numerator and the denominator.
$\frac{22-18i}{20}$

STEP 6

Now, divide the real part and the imaginary part of the numerator by the denominator separately.
$\frac{22}{20} - \frac{18i}{20}$

STEP 7

implify the fractions.
$1.1 -0.9i$ So, the given complex number $\frac{3-i}{6+4i}$ simplifies to $1.1 -0.9i$ .

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