Question

Solve $\frac{9^{\frac{1}{3}}\times {27}^{\frac{-1}{2}}}{3^{\frac{1}{6}}\times 3^{\frac{-2}{3}}}$

Answer 1

$\frac{9^{\frac{1}{3}}\times {27}^{\frac{-1}{2}}}{3^{\frac{1}{6}}\times 3^{\frac{-2}{3}}}=\frac{(3^2{)}^{\frac{1}{3}}\times (3^3{)}^{\frac{-1}{2}}}{3^{\frac{1}{6}}\times 3^{\frac{-2}{3}}}$

$=\frac{3^{\frac{2}{3}}\times 3^{\frac{-3}{2}}}{3^{\frac{1}{6}}\times 3^{\frac{-2}{3}}}\left(as\right(a^m{)}^n=a^{mn})$

$=\frac{3^{\frac{2}{3}-\frac{3}{2}}}{3^{\frac{1}{6}-\frac{2}{3}}}(as{a}^m\times a^n=a^{m+n})$

$=\frac{3^{\frac{-5}{6}}}{3^{\frac{-1}{2}}}=3^{\frac{-5}{6}-\left(\frac{-1}{2}\right)}(as\frac{a^m}{a^n}=a^{m-n})$

$=3^{\frac{1}{2}-\frac{5}{6}}=3^{\frac{-1}{3}}$