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To solve this division of complex numbers, we are given:

\[
\frac{12 - 7i}{6i}
\]

### Step-by-Step Solution

1. **Rewrite the Division as a Multiplication Problem**:
   To eliminate the imaginary part in the denominator, we can multiply both the numerator and the denominator by the complex conjugate of the denominator. In this case, the denominator is \(6i\), and its conjugate is \(-6i\).
   
   \[
   \frac{12 - 7i}{6i} \times \frac{-6i}{-6i} = \frac{(12 - 7i)(-6i)}{(6i)(-6i)}
   \]

2. **Multiply the Numerator**:
   Expand \( (12 - 7i)(-6i) \):
   - \(12 \cdot (-6i) = -72i\)
   - \(-7i \cdot (-6i) = 42i^2\)
   
   Since \(i^2 = -1\), \(42i^2 = 42 \times (-1) = -42\).

   So, the numerator becomes:
   
   \[
   -72i - 42
   \]

   Which can be rewritten as:
   
   \[
   -42 - 72i
   \]

3. **Simplify the Denominator**:
   \((6i)(-6i) = -36i^2 = -36 \times (-1) = 36\)

   So, the expression now is:
   
   \[
   \frac{-42 - 72i}{36}
   \]

4. **Divide Each Term by 36**:
   Simplify by dividing both the real and imaginary parts by 36:

   \[
   \frac{-42}{36} + \frac{-72i}{36} = -\frac{7}{6} - 2i
   \]

### Final Answer
\[
-\frac{7}{6} - 2i
\]

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Would you like more details on any step, or do you have additional questions?

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### Related Questions:
1. How do you find the conjugate of a complex number?
2. Why is it necessary to multiply by the conjugate in complex division?
3. Can you explain the meaning of \(i^2 = -1\) in complex numbers?
4. How would you handle a complex division if both parts of the denominator were non-zero (e.g., \(a + bi\))?
5. What are some practical applications of complex number division?

### Tip:
When dividing complex numbers, always try to eliminate the imaginary unit \(i\) from the denominator by multiplying with the conjugate. This makes the expression easier to work with.

Divide \( \frac{12 - 7i}{6i} \). Simplify your answer in the form \( a + bi \) using integers or fractions.

Learn how to divide complex numbers with a step-by-step solution to \( \frac{12 - 7i}{6i} \). This problem covers key concepts in complex number division, including multiplying by the conjugate, simplifying imaginary units, and expressing the answer in \( a + bi \) form. Suitable for Grades 11-12.

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