Given,

$${{{{(1 + 2i)}^8}\,.\,{{(1 - 2i)}^2}} \over {(3 + 2i)\,.\,\overline {(4 - 6i)} }}$$

$$ = {{{{(1 + 2i)}^2}{{(1 - 2i)}^2}{{(1 + 2i)}^6}} \over {(3 + 2i)(4 + 6i)}}$$

$$ = {{{{(1 - 4{i^2})}^2}{{(1 + 2i)}^6}} \over {12 + 18i + 8i + 12{i^2}}}$$

$$ = {{{{(1 + 5)}^2}{{\left[ {{{(1 + 2i)}^2}} \right]}^3}} \over {12 + 26i - 12}}$$

$$ = {{25{{(1 + 4{i^2} + 4i)}^3}} \over {26i}}$$

$$ = {{25{{(1 - 4 + 4i)}^3}} \over {26i}}$$

$$ = {{25{{( - 3 + 4i)}^3}} \over {26i}}$$

$$ = {{25} \over {26i}}\left[ {{{( - 3)}^3} + {{(4i)}^3} + 3\,.\,{{( - 3)}^2}\,.\,4i + 3( - 3)\,.\,{{(4i)}^2}} \right]$$

$$ = {{25} \over {26i}}( - 27 - 64i + 108i + 144)$$

$$ = {{25} \over {26i}}(117 + 44i)$$

$$ = {{25i} \over {26{i^2}}}(117 + 44i)$$

$$ = {{25i} \over { - 26}}(117 + 44i)$$

$$ = {{25 \times 117i} \over { - 26}} - {{25 \times 44{i^2}} \over {26}}$$

$$ = {{25 \times 117i} \over { - 26}} + {{22 \times 25} \over {13}}$$

$$ = {{25 \times 117i} \over { - 26}} + {{550} \over {13}}$$

$$\therefore$$ Real part $$ = {{550} \over {13}}$$