Question

find the quotient. express your answer in standard form. \\(\frac{-5 - 6i}{-6 - 5i}\\)

Explanation:

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

The conjugate of \(-6 - 5i\) is \(-6 + 5i\). So we have:
\[
\frac{-5 - 6i}{-6 - 5i} \times \frac{-6 + 5i}{-6 + 5i}
\]

Step2: Expand the numerator and the denominator.

First, expand the numerator \((-5 - 6i)(-6 + 5i)\):
\[

$$\begin{align*} (-5)(-6)+(-5)(5i)+(-6i)(-6)+(-6i)(5i)&= 30 - 25i + 36i - 30i^2\\ &= 30 + 11i - 30(-1) \quad (\text{since } i^2 = -1)\\ &= 30 + 11i + 30\\ &= 60 + 11i \end{align*}$$

\]
Next, expand the denominator \((-6 - 5i)(-6 + 5i)\) using the difference of squares formula \((a - b)(a + b)=a^2 - b^2\), where \(a=-6\) and \(b = 5i\):
\[

$$\begin{align*} (-6)^2-(5i)^2&= 36 - 25i^2\\ &= 36 - 25(-1)\\ &= 36 + 25\\ &= 61 \end{align*}$$

\]

Step3: Write the result as a complex number in standard form.

Now we have \(\frac{60 + 11i}{61}\), which can be written as \(\frac{60}{61}+\frac{11}{61}i\).

Answer:

\(\frac{60}{61}+\frac{11}{61}i\)