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$\big[\big(\frac{5}{9}\big)^4÷\big(-\frac{5}{9}\big)^2\big]^2÷\big(\frac{10}{9}\big)^4+\frac{9}{16}=$
$=\big[\big(-\frac{5}{9}\big)^{4-2}\big]^2÷\big(\frac{10}{9}\big)^4+\frac{9}{16}=$
$=\big[\big(-\frac{5}{9}\big)^2\big]^2÷\big(\frac{10}{9}\big)^4+\frac{9}{16}=$
$=\big[\frac{25}{81}\big]^2÷\big(\frac{10}{9}\big)^4+\frac{9}{16}=$
$=\frac{625}{6561}÷\frac{10000}{6561}+\frac{9}{16}=$
$=\frac{625}{6561}×\frac{6561}{10000}+\frac{9}{16}=$
$=\frac{1}{1}×\frac{1}{16}+\frac{9}{16}=$
$=\frac{1}{16}+\frac{9}{16}=$
$=\frac{10}{16}=$
$=\frac{5}{8}$