Question

divide.
\\(\frac{3 + 8i}{2 + 3i}\\)
\\(\frac{3 + 8i}{2 + 3i} = \square\\)
(simplify your answer. use integers or fractions for any numbers in the ex

Explanation:

Step1: Multiply numerator and denominator by the conjugate of the denominator.

The conjugate of \(2 + 3i\) is \(2 - 3i\). So we have:
\[
\frac{3 + 8i}{2 + 3i} \times \frac{2 - 3i}{2 - 3i}
\]

Step2: Expand the numerator and the denominator.

For the numerator: \((3 + 8i)(2 - 3i)=3\times2+3\times(-3i)+8i\times2+8i\times(-3i)=6 - 9i + 16i - 24i^{2}\)
Since \(i^{2}=-1\), this becomes \(6 + 7i - 24\times(-1)=6 + 7i + 24 = 30 + 7i\)
For the denominator: \((2 + 3i)(2 - 3i)=2^{2}-(3i)^{2}=4 - 9i^{2}=4 - 9\times(-1)=4 + 9 = 13\)

Step3: Write the result as a complex number.

\[
\frac{30 + 7i}{13}=\frac{30}{13}+\frac{7}{13}i
\]

Answer:

\(\frac{30}{13}+\frac{7}{13}i\)