\(0 . \overline{7}\) is multiplied by itself and that result is divided by \(\left(\frac{2}{9}\right)^{t h}\) part of the reciprocal of \(0 . \overline{8}\). If the value thus obtained is x, then 9√x =

Given:

Expression to evaluate: \(0 . \overline{7}\) multiplied by itself, then divided by \(\left(\frac{2}{9}\right)^{t h}\) part of the reciprocal of \(0 . \overline{8}\).

Find: \(9\sqrt{x}\)

Formula used:

Conversion of recurring decimal: \(0.\overline{a} = \frac{a}{9}\)

Reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\)

Calculations:

⇒ Convert \(0.\overline{7}\) to fraction: \(0.\overline{7} = \frac{7}{9}\)

⇒ Convert \(0.\overline{8}\) to fraction: \(0.\overline{8} = \frac{8}{9}\)

⇒ \(0 . \overline{7}\) multiplied by itself: \(\left(\frac{7}{9}\right) \times \left(\frac{7}{9}\right) = \frac{49}{81}\)

⇒ Reciprocal of \(0.\overline{8}\): Reciprocal of \(\frac{8}{9}\) is \(\frac{9}{8}\)

⇒ \(\left(\frac{2}{9}\right)^{t h}\) part of the reciprocal of \(0 . \overline{8}\): \(\frac{2}{9} \times \frac{9}{8} = \frac{2}{8} = \frac{1}{4}\)

⇒ Value of x: \(x = \frac{\text{result of step 2}}{\text{result of step 4}} = \frac{\frac{49}{81}}{\frac{1}{4}}\)

⇒ \(x = \frac{49}{81} \times 4 = \frac{196}{81}\)

⇒ Calculate \(9\sqrt{x}\): \(9\sqrt{\frac{196}{81}}\)

⇒ \(9 \times \frac{\sqrt{196}}{\sqrt{81}}\)

⇒ \(9 \times \frac{14}{9}\)

⇒ 14

∴ The value of \(9\sqrt{x}\) is 14.