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):

$$
LATEXBLOCK0
$$

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

$$
LATEXBLOCK1
$$

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

$$
LATEXBLOCK2
$$

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

$$
LATEXBLOCK3
$$

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 \(\boldsymbol{\frac{-28 + 41i}{85}}\))