$$1\frac{5}{9}-[3\frac{1}{9}\div\{4\times2-(3\frac{1}{18}-2\frac{1}{27})\}]$$
$\frac{3766}{3393}\approx 1.109932213$
$$\frac{9+5}{9}-\frac{\frac{3\times 9+1}{9}}{4\times 2-\left(\frac{3\times 18+1}{18}-\frac{2\times 27+1}{27}\right)}$$
$$\frac{14}{9}-\frac{\frac{3\times 9+1}{9}}{4\times 2-\left(\frac{3\times 18+1}{18}-\frac{2\times 27+1}{27}\right)}$$
$$\frac{14}{9}-\frac{\frac{27+1}{9}}{4\times 2-\left(\frac{3\times 18+1}{18}-\frac{2\times 27+1}{27}\right)}$$
$$\frac{14}{9}-\frac{\frac{28}{9}}{4\times 2-\left(\frac{3\times 18+1}{18}-\frac{2\times 27+1}{27}\right)}$$
$$\frac{14}{9}-\frac{\frac{28}{9}}{8-\left(\frac{3\times 18+1}{18}-\frac{2\times 27+1}{27}\right)}$$
$$\frac{14}{9}-\frac{\frac{28}{9}}{8-\left(\frac{54+1}{18}-\frac{2\times 27+1}{27}\right)}$$
$$\frac{14}{9}-\frac{\frac{28}{9}}{8-\left(\frac{55}{18}-\frac{2\times 27+1}{27}\right)}$$
$$\frac{14}{9}-\frac{\frac{28}{9}}{8-\left(\frac{55}{18}-\frac{54+1}{27}\right)}$$
$$\frac{14}{9}-\frac{\frac{28}{9}}{8-\left(\frac{55}{18}-\frac{55}{27}\right)}$$
$$\frac{14}{9}-\frac{\frac{28}{9}}{8-\left(\frac{165}{54}-\frac{110}{54}\right)}$$
$$\frac{14}{9}-\frac{\frac{28}{9}}{8-\frac{165-110}{54}}$$
$$\frac{14}{9}-\frac{\frac{28}{9}}{8-\frac{55}{54}}$$
$$\frac{14}{9}-\frac{\frac{28}{9}}{\frac{432}{54}-\frac{55}{54}}$$
$$\frac{14}{9}-\frac{\frac{28}{9}}{\frac{432-55}{54}}$$
$$\frac{14}{9}-\frac{\frac{28}{9}}{\frac{377}{54}}$$
$$\frac{14}{9}-\frac{28}{9}\times \frac{54}{377}$$
$$\frac{14}{9}-\frac{28\times 54}{9\times 377}$$
$$\frac{14}{9}-\frac{1512}{3393}$$
$$\frac{14}{9}-\frac{168}{377}$$
$$\frac{5278}{3393}-\frac{1512}{3393}$$
$$\frac{5278-1512}{3393}$$
$$\frac{3766}{3393}$$
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$\frac{2 \cdot 7 \cdot 269}{3 ^ {2} \cdot 13 \cdot 29} = 1\frac{373}{3393} = 1.1099322133804892$
