Question

write the quotient in standard form. (simplify your answer completely.)
$\frac{7 - 11i}{1 - 3i}$

Explanation:

Step1: Multiply by conjugate

Multiply numerator and denominator by the conjugate of the denominator $1 + 3i$.
$\frac{(7 - 11i)(1 + 3i)}{(1 - 3i)(1 + 3i)}$

Step2: Expand numerator

Use FOIL method: $(7 - 11i)(1 + 3i)=7\times1+7\times3i-11i\times1 - 11i\times3i=7 + 21i-11i-33i^{2}$. Since $i^{2}=-1$, it becomes $7 + 21i-11i + 33=40 + 10i$.

Step3: Expand denominator

Use the difference - of - squares formula $(a - b)(a + b)=a^{2}-b^{2}$. Here $a = 1$ and $b = 3i$, so $(1 - 3i)(1 + 3i)=1^{2}-(3i)^{2}=1-9i^{2}=1 + 9 = 10$.

Step4: Simplify the fraction

$\frac{40 + 10i}{10}=\frac{40}{10}+\frac{10i}{10}=4 + i$.

Answer:

$4 + i$