Mathematics - Binomial Theorem Question with Solution | TestHub

Suppose $x=(5\sqrt{3}+8{)}^4$ $\begin{array}{rlrl}& =[x]+\{x\}& 0<5\sqrt{3}-8<1& 0<(5\sqrt{3}-8{)}^4<1\end{array}$ Let $F=(5\sqrt{3}-8{)}^4$ $\begin{array}{rlrlrlr}& 0<F<1& [x]+\{x\}+F=(5\sqrt{3}+8{)}^4+(5\sqrt{3}-8{)}^4& ={}^4{C}_0(5\sqrt{3}{)}^4+{}^4{C}_1(5\sqrt{3}{)}^3(8)+\dots .+\left({}^4{C}_0(5\sqrt{3}{)}^4-{}^4{C}_1(5\sqrt{3}{)}^3(8)+\dots \right)& =2.\left[{}^4{C}_0(5\sqrt{3}{)}^4+{}^4{C}_2(8{)}^2(5\sqrt{3}{)}^2+{}^4{C}_4(8{)}^4\right]& =2[625(9)+(6)64(25)(3)+(64)(64)]& =2[5625+28800+4096]=77042\end{array}$ $\{x\}+F$ must be an integer Also $0<\{x\}+F<2$ $\begin{array}{rlr}& \Rightarrow \{x\}+F=1& \Rightarrow [x]=77042-1=77041\end{array}$