Question

Divide and simplify
(-86 - 72i) ÷ (-6-7i)=

Answer 1

To divide the complex numbers \((-86 - 72i)\) by \((-6 - 7i)\), we can multiply the numerator and denominator by the conjugate of the denominator. The conjugate of \((-6 - 7i)\) is \((-6 + 7i)\).

Let's perform the division:

\[
\frac{-86 - 72i}{-6 - 7i} \cdot \frac{-6 + 7i}{-6 + 7i} = \frac{(-86 - 72i)(-6 + 7i)}{(-6 - 7i)(-6 + 7i)}
\]

Now we will calculate the denominator first:

\[
(-6 - 7i)(-6 + 7i) = (-6)^2 - (7i)^2 = 36 - 49(-1) = 36 + 49 = 85
\]

Next, we calculate the numerator:

\[
(-86 - 72i)(-6 + 7i) = -86 \cdot -6 + (-86) \cdot 7i - 72i \cdot -6 + (-72i) \cdot 7i
\]

Calculating each term, we have:

1. \(-86 \cdot -6 = 516\)
2. \(-86 \cdot 7i = -602i\)
3. \(-72i \cdot -6 = 432i\)
4. \(-72i \cdot 7i = -504i^2 = 504\) (since \(i^2 = -1\))

Now, combining all these:

\[
516 + 504 + (-602i + 432i) = (516 + 504) + (-602 + 432)i = 1020 - 170i
\]

Thus, the numerator is \(1020 - 170i\). Now we can combine this with the denominator:

\[
\frac{1020 - 170i}{85}
\]

Now, we'll separate the real and imaginary parts:

\[
= \frac{1020}{85} - \frac{170i}{85} = 12 - 2i
\]

So the simplified result of \((-86 - 72i) ÷ (-6 - 7i)\) is:

\[
\boxed{12 - 2i}
\]