Chapter A.1, Problem 52E

Solution

Write using a single radical.

  ${(3x)}^{1/6}$ .

Given:

The given expression is $\sqrt{\sqrt[3]{3x^2}}$ .

Concept Used:

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

Calculation:

The given expression is $\sqrt{\sqrt[3]{3x^2}}$ .

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

Thus the single radical expression of $\sqrt{\sqrt[3]{3x^2}}$ is ${(3x)}^{1/6}$ .