Question

simplify: \\(\frac{3 + 7i}{7 - 11i}\\)\
\\(\bigcirc\\) the expression is already simplified\
\\(\bigcirc\\) no solutions\
\\(\bigcirc\\) \\(\frac{-28 + 41i}{85}\\)\
\\(\bigcirc\\) \\(\frac{14i}{-4i}\\)

Explanation:

Step1: Multiply numerator and denominator by conjugate of denominator

The conjugate of \(7 - 11i\) is \(7 + 11i\). So we multiply the fraction \(\frac{3 + 7i}{7 - 11i}\) by \(\frac{7 + 11i}{7 + 11i}\).
\[
\frac{(3 + 7i)(7 + 11i)}{(7 - 11i)(7 + 11i)}
\]

Step2: Expand numerator and denominator

First, expand the numerator using the distributive property (FOIL method):
\[

$$\begin{align*} (3 + 7i)(7 + 11i)&=3\times7+3\times11i + 7i\times7+7i\times11i\\ &=21 + 33i+49i + 77i^{2} \end{align*}$$

\]
Since \(i^{2}=- 1\), we have:
\[

$$\begin{align*} 21 + 33i+49i + 77i^{2}&=21+(33i + 49i)+77\times(-1)\\ &=21 + 82i-77\\ &=- 56+82i \end{align*}$$

\]
Now, expand the denominator using the difference of squares formula \((a - b)(a + b)=a^{2}-b^{2}\), where \(a = 7\) and \(b = 11i\):
\[

$$\begin{align*} (7 - 11i)(7 + 11i)&=7^{2}-(11i)^{2}\\ &=49-121i^{2} \end{align*}$$

\]
Again, since \(i^{2}=-1\), we get:
\[

$$\begin{align*} 49-121i^{2}&=49-121\times(-1)\\ &=49 + 121\\ &=170 \end{align*}$$

\]
So the fraction becomes \(\frac{-56 + 82i}{170}\)

Step3: Simplify the fraction

We can factor out a 2 from the numerator:
\[
\frac{2(-28 + 41i)}{170}=\frac{-28 + 41i}{85}
\]

Answer:

\(\frac{-28 + 41i}{85}\) (corresponding to the option \(\boldsymbol{\frac{-28 + 41i}{85}}\))