\frac{-4-6i}{-7-3i}          Write your answer in a+b⁢i form

To express \(\frac{-4-6i}{-7-3i}\) in the form \(a + bi\), we'll multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(-7 - 3i\) is \(-7 + 3i\).

So, we multiply as follows:

\[
\frac{-4-6i}{-7-3i} \times \frac{-7+3i}{-7+3i} = \frac{(-4-6i)(-7+3i)}{(-7-3i)(-7+3i)}
\]

Let's calculate the denominator first:

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

Since \(i^2 = -1\), the denominator simplifies to:

\[
49 - 9(-1) = 49 + 9 = 58
\]

Now, calculate the numerator using the distributive property (FOIL method):

\[
(-4-6i)(-7+3i) = (-4)(-7) + (-4)(3i) + (-6i)(-7) + (-6i)(3i)
\]

Calculate each term:

1. \((-4)(-7) = 28\)
2. \((-4)(3i) = -12i\)
3. \((-6i)(-7) = 42i\)
4. \((-6i)(3i) = -18i^2\)

Since \(i^2 = -1\), \(-18i^2\) simplifies to \(18\). Now, combine all the terms:

\[
28 - 12i + 42i + 18 = 28 + 18 + (-12i + 42i) = 46 + 30i
\]

Now, put it all together:

\[
\frac{46 + 30i}{58}
\]

This simplifies by dividing both the real and imaginary parts by 58:

\[
\frac{46}{58} + \frac{30i}{58} = \frac{23}{29} + \frac{15}{29}i
\]

So the final answer is:

\[
\boxed{\frac{23}{29} + \frac{15}{29}i}
\]

Would you like any further explanation or details? Here are some related problems you might find useful:

1. Simplify \(\frac{3 + 4i}{1 - 2i}\) and write in \(a + bi\) form.
2. Express \(\frac{7 - 9i}{2 + 3i}\) in the form \(a + bi\).
3. Find the value of \(\frac{5 - 2i}{4 + i}\) and write it as a complex number.
4. Simplify \(\frac{8 + 6i}{3 - 4i}\) and write in \(a + bi\) form.
5. Perform the division \(\frac{-1 + 2i}{-3 + 5i}\) and simplify the result.

**Tip:** Multiplying by the conjugate is a key technique to eliminate the imaginary part from the denominator.

Learn how to simplify the division of complex numbers (-4-6i)/(-7-3i) and express the result in the form a + bi. This problem involves using the conjugate of the denominator and performing complex number division. Suitable for advanced high school students studying complex numbers.

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