$3^{2n }= 9^n= ( 1 + 8{)}^n$
 $\Rightarrow 3^{2n}{=}^n{C}_0 {+}^n{C}_1·8{+}^n{C}_2·8^2+.....{+}^n{C}_n·8^n$

$\Rightarrow \frac{3^{2n}}{8}=\frac{{}^n{C}_0 {+}^n{C}_1·8{+}^n{C}_2·8^2+.....{+}^n{C}_n·8^n}{8}$
$\Rightarrow \frac{3^{2n}}{8}=\frac{1}{8}+\left({}^n{C}_1 {+}^n{C}_2·8 +......+{ }^n{C}_n·8^{n - 1 }\right)$
 $\Rightarrow \frac{3^{2n}}{8}=\frac{1}{8}$ $+$ integer quantity

$\left\{x\right\}\in [0, 1) \& \left\{\mathrm{I}\right\}=0$
$\Rightarrow$    $\left\{\frac{3^{2n}}{8}\right\}=\frac{1}{8}+0$