Chapter 6.6, Problem 34PPS

To determine

Simplify the expression $\frac{\sqrt[8]{81}}{\sqrt[6]{3}}$ .

Answer to Problem 34PPS

The simplified value of expression $\frac{\sqrt[8]{81}}{\sqrt[6]{3}}$ is $3^{\frac{1}{3}}$ .

Explanation of Solution

Given information: The given expression is $\frac{\sqrt[8]{81}}{\sqrt[6]{3}}$ .

Calculation: The given expression $\frac{\sqrt[8]{81}}{\sqrt[6]{3}}$ can be simplified as,

  $\begin{array}{c}\frac{\sqrt[8]{81}}{\sqrt[6]{3}}=\frac{{\left(81\right)}^{\frac{1}{8}}}{{\left(3\right)}^{\frac{1}{6}}}\mathrm{......}\left(\text{Using }\sqrt[n]{a}={\left(a\right)}^{\frac{1}{n}}\right)\\ =\frac{{\left(3^4\right)}^{\frac{1}{8}}}{{\left(3\right)}^{\frac{1}{6}}}\mathrm{......}\left(\text{Factorizing 81=}{\left(\text{3}\right)}^4\right)\\ =\frac{3^{\frac{1}{2}}}{3^{\frac{1}{6}}}\\ ={\left(3\right)}^{\frac{1}{2}-\frac{1}{6}}\mathrm{....}\left(\text{Using }\frac{x^m}{x^n}=x^{m-n}\right)\\ =3^{\frac{1}{3}}\end{array}$