QuestionEvaluate $\frac{3-4 i}{5 i}$ and express the result as $a + b i$ .

Studdy Solution

STEP 1

Assumptions1. The expression to be evaluated is $\frac{3-4 i}{5 i}$ . The result should be in the form $a+bi$ , where $a$ and $b$ are real numbers and $i$ is the imaginary unit with the property $i^ = -1$ .

STEP 2

To simplify the expression, we can multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number $a+bi$ is $a-bi$ . In this case, the conjugate of $5i$ is $-5i$ .
$\frac{-4 i}{5 i} \times \frac{-5i}{-5i}$

STEP 3

Multiply the numerators together.
$(3-i) \times -5i = -15i +20i^2$

STEP 4

Multiply the denominators together.
$i \times -i = -25i^2$

STEP 5

Substitute $i^2$ with $-1$ in the numerator and denominator.
umerator $-15i +20(-1) = -15i -20$
Denominator $-25(-1) =25$

STEP 6

The expression now becomes $\frac{-15i -20}{25}$ , which can be written as $\frac{-20}{25} + \frac{-15i}{25}$ .

STEP 7

implify the real and imaginary parts separately.
Real part $\frac{-20}{25} = -0.$
Imaginary part $\frac{-15i}{25} = -0.6i$

STEP 8

The simplified expression in the form $a+bi$ is $-0.8 -0.6i$ .
The expression $\frac{3-4 i}{5 i}$ evaluates to $-0.8 -0.6i$ .

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