Ta có: \(0,\left(2\right)=2.0,\left(1\right)=2.\frac{1}{9}=\frac{2}{9};\)
\(1,8\left(2\right)=1,8+0,0\left(2\right)=1,8+\frac{1}{10}.0,\left(2\right)=1,8+\frac{1}{10}.2.0,\left(1\right)=1,8+\frac{1}{10}.2.\frac{1}{9}=\frac{82}{45}\)
\(\Rightarrow0,\left(2\right)+1,8\left(2\right)=\frac{2}{9}+\frac{82}{45}=\frac{92}{45}\)
\(0,\left(2\right)+1,8\left(2\right)\)
=\(0,\left(1\right).2+1,8+0,0\left(2\right)\)
=\(\frac{1}{9}.2+1,8+0,0\left(1\right).2\)
=\(\frac{2}{9}+\frac{9}{5}+\frac{1}{90}.2\)
=\(\frac{20}{90}+\frac{162}{90}+\frac{2}{90}\)
=\(\frac{184}{90}\)
=\(\frac{92}{45}\)