I have the question: Evaluate
$$\frac{9^{1/3} \cdot 27^{-1/2}}{3^{-1/6} \cdot 3^{-2/3}}$$
I have use the rules of indices to get (cuberoot 9 X 1/squareroot 27) / (sixthroot 3 X 1/cuberoot 3^2) and the answer I get is 0.2.
However, the solutions say that the answer should be 1 and I do not understand why.
You have
\begin{align} \frac{9^{1/3} \cdot 27^{-1/2}}{3^{-1/6} \cdot 3^{-2/3}} =\frac{(3^2)^{1/3} \cdot (3^3)^{-1/2}}{3^{-1/6-2/3} }=\frac{3^{2/3} \cdot 3^{-3/2}}{3^{-5/6} }=3^{2/3-3/2}\cdot 3^{5/6}=3^{-5/6}\cdot 3^{5/6}=1. \end{align}