Question

Value of $\sqrt[8]{8(625{)}^{0.\overline{3}}\times (625{)}^{0.\overline{6}}\times (625{)}^{0.\overline{9}}}$ is equal to

• A.
  $5$
• B. $25$
• C. $125$
• D. $625$

Answer 1

Answer: (A)

$5$

$\sqrt[8]{8{\left(5^4\right)}^{0.\overline{3}}\times {\left(5^4\right)}^{0.\overline{6}}\times {\left(5^4\right)}^{0.\overline{9}}}$

We have $0.\overline{3}+0.\overline{6}+0.\overline{9}=\mathrm{0.33333...}+\mathrm{0.66666...}+\mathrm{0.99999...}=1.99999=1.\overline{9}\approx 2$

$=\sqrt[8]{8{\left(5^4\right)}^{0.\overline{3}+0.\overline{6}+0.\overline{9}}}$

$=\sqrt[8]{8{\left(5^4\right)}^2}$

$=\sqrt[8]{85^8}$

$=\left|5\right|=5$