Questions: Divide and express the result in standard form. 6 i / (4-8 i) 6 i / (4-8 i) = (Simplify your answer. Type your answer in the form a+bi. Use integers or fractions for any numbers in the expression.)

$Divide and express the result in standard form. 6 i / (4-8 i) 6 i / (4-8 i) = (Simplify your answer. Type your answer in the form a+bi. Use integers or fractions for any numbers in the expression.)$

Transcript text: Divide and express the result in standard form. \[ \begin{array}{c} \frac{6 i}{4-8 i} \\ \frac{6 i}{4-8 i}= \end{array} \] $\square$ (Simplify your answer. Type your answer in the form $\mathrm{a}+\mathrm{bi}$. Use integers or fractions for any numbers in the expression.)

Solution

Solution Steps

To divide complex numbers, we multiply the numerator and the denominator by the conjugate of the denominator. This will help us eliminate the imaginary part in the denominator and simplify the expression.

Step 1: Multiply the Numerator and Denominator by the Conjugate of the Denominator

To simplify the division of complex numbers, we multiply both the numerator and the denominator by the conjugate of the denominator: \[ \frac{6i}{4 - 8i} \times \frac{4 + 8i}{4 + 8i} \]

Step 2: Calculate the Numerator

The numerator becomes: \[ 6i \times (4 + 8i) = 6i \times 4 + 6i \times 8i = 24i + 48i^2 \] Since $i^2 = -1$, we have: \[ 24i + 48(-1) = 24i - 48 = -48 + 24i \]

Step 3: Calculate the Denominator

The denominator becomes: \[ (4 - 8i) \times (4 + 8i) = 4^2 - (8i)^2 = 16 - 64i^2 \] Since $i^2 = -1$, we have: \[ 16 - 64(-1) = 16 + 64 = 80 \]

Step 4: Simplify the Expression

Now, we divide the simplified numerator by the simplified denominator: \[ \frac{-48 + 24i}{80} = \frac{-48}{80} + \frac{24i}{80} = -0.6 + 0.3i \]