If `z_(1),z_(2)` are respectively `1-i,-2+4i,` find `I_(m){(z_(1)z_(2))/(z_(1))}`.

Answer

To solve the problem, we need to find the imaginary part of the expression \(\frac{z_1 z_2}{z_1}\), where \(z_1 = 1 - i\) and \(z_2 = -2 + 4i\).

### Step-by-Step Solution:

1. **Identify the Complex Numbers**:
- Let \(z_1 = 1 - i\)
- Let \(z_2 = -2 + 4i\)

2. **Calculate the Product \(z_1 z_2\)**:
\[ z_1 z_2 = (1 - i)(-2 + 4i) \]
Using the distributive property (FOIL method):
\[ z_1 z_2 = 1 \cdot (-2) + 1 \cdot (4i) - i \cdot (-2) - i \cdot (4i) \]
\[ = -2 + 4i + 2i - 4(-1) \]
\[ = -2 + 4i + 2i + 4 \]
\[ = 2 + 6i \]

3. **Divide by \(z_1\)**:
\[ \frac{z_1 z_2}{z_1} = \frac{2 + 6i}{1 - i} \]

4. **Multiply by the Conjugate**:
To simplify the division, multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{(2 + 6i)(1 + i)}{(1 - i)(1 + i)} \]
The denominator simplifies as follows:
\[ (1 - i)(1 + i) = 1^2 - i^2 = 1 - (-1) = 2 \]
Now calculate the numerator:
\[ (2 + 6i)(1 + i) = 2 \cdot 1 + 2 \cdot i + 6i \cdot 1 + 6i \cdot i \]
\[ = 2 + 2i + 6i - 6 \]
\[ = -4 + 8i \]

5. **Combine the Results**:
Now we have:
\[ \frac{-4 + 8i}{2} = -2 + 4i \]

6. **Find the Imaginary Part**:
The imaginary part of \(-2 + 4i\) is \(4\).

### Final Answer:
The imaginary part of \(\frac{z_1 z_2}{z_1}\) is \(4\).