Question

Find the conjugate of Z =(3-2i)(2+3i)/(1+2i)(2-1i)

Answer 1

To find the conjugate of a complex expression, we first need to simplify it. Let's start with the given expression:

\[
Z = \frac{(3 - 2i)(2 + 3i)}{(1 + 2i)(2 - i)}
\]

### Simplifying the Numerator
1. Calculate \((3 - 2i)(2 + 3i)\):
\[
(3 - 2i)(2 + 3i) = 3 \cdot 2 + 3 \cdot 3i - 2i \cdot 2 - 2i \cdot 3i
\]
\[
= 6 + 9i - 4i - 6i^2
\]
Remembering that \(i^2 = -1\), we have:
\[
= 6 + 5i + 6 = 12 + 5i
\]

### Simplifying the Denominator
2. Next, calculate \((1 + 2i)(2 - i)\):
\[
(1 + 2i)(2 - i) = 1 \cdot 2 + 1 \cdot (-i) + 2i \cdot 2 + 2i \cdot (-i)
\]
\[
= 2 - i + 4i - 2i^2
\]
Again using \(i^2 = -1\):
\[
= 2 + 3i + 2 = 4 + 3i
\]

### Full Simplification of Z
Now we substitute back into the expression for \(Z\):
\[
Z = \frac{12 + 5i}{4 + 3i}
\]

### Finding the Conjugate
To find the conjugate of the complex number \(Z\), we take the conjugate of both the numerator and the denominator. The conjugate of \(a + bi\) is \(a - bi\).

Now, we can use the conjugates:
- The conjugate of \(12 + 5i\) is \(12 - 5i\).
- The conjugate of \(4 + 3i\) is \(4 - 3i\).

Thus, the conjugate of \(Z\) is:
\[
\text{Conjugate of } Z = \frac{12 - 5i}{4 - 3i}
\]

### Conclusion
Therefore, the conjugate of \(Z = \frac{(3-2i)(2+3i)}{(1+2i)(2-i)}\) is:

\[
\frac{12 - 5i}{4 - 3i}
\]