Question

Simplify
$\frac{\left(0.3{)}^{\frac{1}{3}}{\left(\frac{1}{27}\right)}^{\frac{1}{4}}\right(9{)}^{\frac{1}{6}}(0.81{)}^{\frac{2}{3}}}{\left(0.9{)}^{\frac{2}{3}}\right(3{)}^{-\frac{1}{2}}{\left(\frac{1}{3}\right)}^{-2}(243{)}^{-\frac{1}{4}}}$

• A. 0.3
• B. 0.9
• C. 1.27
• D. 0.09

Answer 1

The correct option is A

0.3

$\frac{\left(0.3{)}^{\frac{1}{3}}{\left(\frac{1}{27}\right)}^{\frac{1}{4}}\right(9{)}^{\frac{1}{6}}(0.81{)}^{\frac{2}{3}}}{\left(0.9{)}^{\frac{2}{3}}\right(3{)}^{-\frac{1}{2}}{\left(\frac{1}{3}\right)}^{-2}(243{)}^{-\frac{1}{4}}}$
$=\frac{{\left(\frac{3}{10}\right)}^{\frac{1}{3}}{\left(\frac{1}{3}\right)}^{\frac{3}{4}}(3^2{)}^{\frac{1}{6}}{\left(\frac{81}{100}\right)}^{\frac{2}{3}}}{{\left(\frac{9}{10}\right)}^{\frac{2}{3}}\left(3{)}^{-\frac{1}{2}}\right(3^{-1}{)}^{-2}(3^5{)}^{-\frac{1}{4}}}$
$=\frac{\left(10{)}^{-\frac{1}{3}}\right(3{)}^{\frac{1}{3}}\left(3{)}^{-\frac{3}{4}}\right(3{)}^{\frac{1}{3}}\left(3{)}^{4\times \frac{2}{3}}\right({10}^{-2}{)}^{\frac{2}{3}}}{(3{)}^{\frac{4}{3}}\times (10{)}^{-\frac{2}{3}}\times (3{)}^{-\frac{1}{2}}\times (3{)}^2\times (3{)}^{-\frac{5}{4}}}$
$=\frac{\left(3{)}^{\frac{1}{3}-\frac{3}{4}+\frac{1}{3}+\frac{8}{3}}\right(10{)}^{-\frac{4}{3}-\frac{1}{3}}}{(3{)}^{\frac{4}{3}-\frac{1}{2}+2-\frac{5}{4}}\times (10{)}^{-\frac{2}{3}}}$
$=\frac{(3{)}^{\frac{31}{12}}.(10{)}^{-\frac{5}{3}}}{\left(3{)}^{\frac{19}{12}}\right(10{)}^{\frac{2}{3}}}$
$=\left(3{)}^{\frac{31}{12}-\frac{19}{12}}\right(10{)}^{\frac{5}{3}+\frac{2}{3}}=3^1\times {10}^{-1}=\frac{3}{10}=0.3$