$f\left\{p\right\}$ denotes the fractional part of the number $p$, then $\left\{\frac{3^{200}}{8}\right\}$ is equal to 

$\begin{array}{l}{3}^{200}={\left(3^2\right)}^{100}=(1+8{)}^{100}\\ \Rightarrow 3^{200}=1{+}^{100}{C}_1\times 8{+}^{100}{C}_2\times 8^2+\dots {+}^{100}{C}_{100}\times 8^{100}\\ \Rightarrow \frac{3^{200}}{8}=\frac{1}{8}{+}^{100}{C}_1{+}^{100}{C}_2\times 8+\dots {+}^{100}{C}_{100}\times 8^{99}\end{array}$

$\Rightarrow \frac{3^{200}}{8}=\frac{1}{8}+$a natural number 

$\Rightarrow \left\{\frac{3^{200}}{8}\right\}=\frac{1}{8}$