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Question Number 20174 by DKumar last updated on 23/Aug/17

$$\left(\mathrm{0}.\overset{−} {\mathrm{1}}\right)^{\mathrm{2}} \left\{\mathrm{1}−\mathrm{9}\left(\mathrm{0}.\mathrm{1}\overset{−} {\mathrm{6}}\right)^{\mathrm{2}} \right\} \\$$

Answered by mrW1 last updated on 23/Aug/17

$$\mathrm{0}.\overline {\mathrm{1}}=\mathrm{0}.\mathrm{111111}...=\frac{\mathrm{1}}{\mathrm{9}} \\$$ $$\mathrm{0}.\mathrm{1}\overset{−} {\mathrm{6}}=\mathrm{0}.\mathrm{166666}...=\frac{\mathrm{1}}{\mathrm{6}} \\$$ $$\left(\mathrm{0}.\overset{−} {\mathrm{1}}\right)^{\mathrm{2}} \left\{\mathrm{1}−\mathrm{9}\left(\mathrm{0}.\mathrm{1}\overset{−} {\mathrm{6}}\right)^{\mathrm{2}} \right\} \\$$ $$=\left(\frac{\mathrm{1}}{\mathrm{9}}\right)^{\mathrm{2}} \left[\mathrm{1}−\mathrm{3}^{\mathrm{2}} ×\left(\frac{\mathrm{1}}{\mathrm{6}}\right)^{\mathrm{2}} \right] \\$$ $$=\left(\frac{\mathrm{1}}{\mathrm{9}}\right)^{\mathrm{2}} \left[\mathrm{1}−\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} \right] \\$$ $$=\frac{\mathrm{1}}{\mathrm{81}}×\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}\right)\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\right) \\$$ $$=\frac{\mathrm{1}}{\mathrm{81}}×\frac{\mathrm{3}}{\mathrm{2}}×\left(\frac{\mathrm{1}}{\mathrm{2}}\right) \\$$ $$=\frac{\mathrm{1}}{\mathrm{27}}×\frac{\mathrm{1}}{\mathrm{4}} \\$$ $$=\frac{\mathrm{1}}{\mathrm{108}} \\$$

Answered by ajfour last updated on 23/Aug/17

$$\left(\frac{\mathrm{1}}{\mathrm{9}}\right)^{\mathrm{2}} \left\{\mathrm{1}−\mathrm{9}×\frac{\mathrm{1}}{\mathrm{36}}\right\}=\frac{\mathrm{1}}{\mathrm{9}×\mathrm{9}}×\frac{\mathrm{3}}{\mathrm{4}}=\frac{\mathrm{1}}{\mathrm{108}}\:. \\$$