Divide the complex numbers by rationalizing the denominator and combining all like terms.
1
−
6
i
−
5
−
2
i
−5−2i
1−6i
​


To divide the complex numbers \(\frac{1 - 6i}{-5 - 2i}\), we can follow the steps below to rationalize the denominator and simplify the expression:

### Step 1: Multiply both numerator and denominator by the complex conjugate of the denominator.
The complex conjugate of \(-5 - 2i\) is \(-5 + 2i\). Multiplying by the conjugate will eliminate the imaginary part in the denominator.

\[
\frac{1 - 6i}{-5 - 2i} \times \frac{-5 + 2i}{-5 + 2i}
\]

### Step 2: Perform the multiplication for both numerator and denominator.

#### Numerator:
\[
(1 - 6i)(-5 + 2i) = 1(-5) + 1(2i) + (-6i)(-5) + (-6i)(2i)
\]
Breaking this down:
\[
1(-5) = -5, \quad 1(2i) = 2i, \quad (-6i)(-5) = 30i, \quad (-6i)(2i) = -12i^2
\]
Since \(i^2 = -1\), we have:
\[
-12i^2 = 12
\]
So, the numerator becomes:
\[
-5 + 2i + 30i + 12 = 7 + 32i
\]

#### Denominator:
\[
(-5 - 2i)(-5 + 2i) = (-5)^2 - (2i)^2 = 25 - 4i^2
\]
Since \(i^2 = -1\), this simplifies to:
\[
25 - 4(-1) = 25 + 4 = 29
\]

### Step 3: Write the result.
Now we can write the result as:
\[
\frac{7 + 32i}{29}
\]
This is the simplified form of the division.

### Step 4: Separate into real and imaginary parts (if needed):
\[
\frac{7}{29} + \frac{32}{29}i
\]

Thus, the final result is:
\[
\frac{7}{29} + \frac{32}{29}i
\]

Let me know if you'd like more details or have questions! 

Here are 5 related questions for further practice:
1. Divide \(\frac{3 - 2i}{4 + 3i}\) and simplify.
2. Multiply \((1 + i)(1 - i)\).
3. Rationalize the denominator of \(\frac{2}{1 + 3i}\).
4. Find the modulus of the complex number \(7 + 32i\).
5. Express the complex number \(\frac{5}{i}\) in standard form.

**Tip:** When multiplying complex numbers, remember that \(i^2 = -1\), which helps in simplifying terms containing \(i^2\).

Divide the complex numbers by rationalizing the denominator and combining all like terms: (1 - 6i) / (-5 - 2i).

Learn how to divide complex numbers by rationalizing the denominator in the expression (1 - 6i) / (-5 - 2i). This step-by-step solution covers the use of complex conjugates and imaginary number properties, making it perfect for Grades 11-12 students.

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