Question

Divide and simplify
(-111-27i) ÷ (-7+3i)=

Answer 1

To divide and simplify \((-111 - 27i) \div (-7 + 3i)\), we can multiply the numerator and the denominator by the conjugate of the denominator.

The conjugate of \(-7 + 3i\) is \(-7 - 3i\).

Let's perform the division:

\[
\frac{-111 - 27i}{-7 + 3i} \cdot \frac{-7 - 3i}{-7 - 3i}
\]

Now we will calculate the numerator and the denominator separately.

**Step 1: Calculate the numerator:**

\[
(-111 - 27i)(-7 - 3i) = -111 \cdot -7 + (-111)(-3i) + (-27i)(-7) + (-27i)(-3i)
\]

Calculating each term:

1. \(-111 \cdot -7 = 777\)
2. \(-111 \cdot -3i = 333i\)
3. \(-27i \cdot -7 = 189i\)
4. \(-27i \cdot -3i = 81\) (since \(i^2 = -1\))

Now, combining these:

\[
777 + 333i + 189i + 81 = 777 + 81 + (333 + 189)i = 858 + 522i
\]

**Step 2: Calculate the denominator:**

\[
(-7 + 3i)(-7 - 3i) = (-7)^2 + (3i)^2 = 49 + 9 = 58
\]

**Putting it all together:**

Now we have:

\[
\frac{858 + 522i}{58}
\]

**Step 3: Simplify:**

To simplify, we can divide both the real and imaginary parts by \(58\):

\[
\frac{858}{58} + \frac{522i}{58}
\]

Calculating each:

1. \(\frac{858}{58} = 14.7931\) (approximately) or exactly \(\frac{429}{29}\)
2. \(\frac{522}{58} = 9\)

Thus, the result is:

\[
\frac{429}{29} + 9i
\]

So the simplified form of \((-111 - 27i) \div (-7 + 3i)\) is:

\[
\frac{429}{29} + 9i
\]