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 \((-5 - 6i)(-6 + 5i)\):

$$
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$$

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 \(b = 5i\):

$$
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$$

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:

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