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The problem asks you to simplify the expression:

\[
\frac{1 + 6i}{2 - 4i}
\]

### Steps to Simplify:
1. **Multiply both numerator and denominator by the conjugate** of the denominator. The conjugate of \( 2 - 4i \) is \( 2 + 4i \).
   
   Multiply both the numerator and the denominator by \( 2 + 4i \):
   \[
   \frac{(1 + 6i)(2 + 4i)}{(2 - 4i)(2 + 4i)}
   \]

2. **Simplify the denominator**:
   \[
   (2 - 4i)(2 + 4i) = 2^2 - (4i)^2 = 4 - (-16) = 4 + 16 = 20
   \]
   So, the denominator is 20.

3. **Simplify the numerator**:
   \[
   (1 + 6i)(2 + 4i) = 1 \cdot 2 + 1 \cdot 4i + 6i \cdot 2 + 6i \cdot 4i
   \]
   \[
   = 2 + 4i + 12i + 24i^2
   \]
   Recall that \( i^2 = -1 \), so:
   \[
   24i^2 = 24 \times (-1) = -24
   \]
   Therefore, the numerator becomes:
   \[
   2 + 4i + 12i - 24 = -22 + 16i
   \]

4. **Final expression**:
   Now divide both the real and imaginary parts by 20:
   \[
   \frac{-22 + 16i}{20} = \frac{-22}{20} + \frac{16i}{20} = -\frac{11}{10} + \frac{4i}{5}
   \]

So, the simplified expression is:

\[
-\frac{11}{10} + \frac{4i}{5}
\]

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Would you like further details on any part of this solution?

Here are some related questions:
1. How do you find the conjugate of a complex number?
2. What is the significance of multiplying by the conjugate?
3. Why does \( i^2 \) equal -1?
4. How would the process differ if the denominator had been a purely real number?
5. How can complex fractions be simplified differently?

**Tip:** Always remember to express your final answer in the standard form \( a + bi \), where \( a \) and \( b \) are real numbers.

Simplify the following expression: (1 + 6i) / (2 - 4i)

Learn how to simplify complex fractions like (1 + 6i) / (2 - 4i) by using the conjugate method. This detailed step-by-step solution is suitable for high school students and covers key concepts like complex numbers, algebraic simplifications, and conjugates.

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