Find the conjugate of the following complex number:

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

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#### Solution

\[\text { Let } z = \frac{\left( 3 - 2i \right)\left( 2 + 3i \right)}{\left( 1 + 2i \right)\left( 2 - i \right)}\]

\[ = \frac{6 + 9i - 4i - 6 i^2}{2 - i + 4i - 2 i^2}\]

\[ = \frac{6 + 6 + 5i}{2 + 2 + 3i}\]

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

\[ = \frac{48 - 36i + 20i - 15 i^2}{16 - 9 i^2}\]

\[ = \frac{63 - 16i}{25}\]

`therefore overlineZ =(63 +16i)/25`

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