Question

Find the conjugate of Z is equal to(3-2i)(2+3i)/1+2i)(2-1i)

Answer 1

To find the conjugate of the expression \( Z = \frac{(3 - 2i)(2 + 3i)}{(1 + 2i)(2 - i)} \), we can follow these steps:

1. **Calculate the numerator and the denominator separately.**
2. **Find the conjugate of the resulting complex number.**

### Step 1: Simplify the numerator

**Numerator:**
\[
(3 - 2i)(2 + 3i) = 3 \cdot 2 + 3 \cdot 3i - 2i \cdot 2 - 2i \cdot 3i
\]
\[
= 6 + 9i - 4i + 6 = 12 + 5i
\]

### Step 2: Simplify the denominator

**Denominator:**
\[
(1 + 2i)(2 - i) = 1 \cdot 2 + 1 \cdot (-i) + 2i \cdot 2 + 2i \cdot (-i)
\]
\[
= 2 - i + 4i - 2 = 0 + 3i = 3i
\]

### Complete the expression
So we have:

\[
Z = \frac{12 + 5i}{3i}
\]

### Step 3: Simplify Z
To simplify, we can multiply both the numerator and the denominator by \( -i \) (the conjugate of \( 3i \)) to make the denominator real:

\[
Z = \frac{(12 + 5i)(-i)}{3i \cdot (-i)} = \frac{(-12i - 5)}{3} = \frac{-5 - 12i}{3} = -\frac{5}{3} - 4i
\]

### Step 4: Finding the conjugate of Z
The conjugate of a complex number \( a + bi \) is \( a - bi \). Thus, the conjugate of \( Z = -\frac{5}{3} - 4i \) is:

\[
-\frac{5}{3} + 4i
\]

### Final answer
The conjugate of \( Z \) is:

\[
-\frac{5}{3} + 4i
\]