Chapter A.1, Problem 54E

Solution

Write using a single radical.

  ${\left(ab\right)}^{\frac{1}{6}}$ .

Given:

The given expression is $\sqrt[3]{\sqrt{ab}}$ .

Concept Used:

• Radical - The v symbol that is used to denote square root or nth roots.

Calculation:

The given expression is $\sqrt[3]{\sqrt{ab}}$ .

  $\begin{array}{c}\sqrt[3]{\sqrt{ab}}=\sqrt[3]{{(ab)}^{1/2}}\\ ={\left({(ab)}^{1/2}\right)}^{1/3}\\ ={\left(ab\right)}^{\frac{1}{2}\cdot \frac{1}{3}}\\ ={\left(ab\right)}^{\frac{1}{6}}\end{array}$

Thus the single radical expression of $\sqrt[3]{\sqrt{ab}}$ is ${\left(ab\right)}^{\frac{1}{6}}$ .