Chapter 10.1, Problem 31OE

To determine

To simplify: the given expression.

Answer to Problem 31OE

  $\frac{\sqrt[3]{4}}{\sqrt[6]{2}}=\sqrt{2}$

Explanation of Solution

Given information:

Given expression is

  $\frac{\sqrt[3]{4}}{\sqrt[6]{2}}$

Calculation:

Consider the given expression,

  $\frac{\sqrt[3]{4}}{\sqrt[6]{2}}$

Now, simplify the expression as shown,

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

Simplified value of the expression is $\sqrt{2}$ .