Chapter 7.3, Problem 35PPS

To determine

To simplify ${\left(\frac{1}{81}\right)}^{\frac{1}{4}}.$

Answer to Problem 35PPS

  $\frac{1}{3}$

Explanation of Solution

Given:

  ${\left(\frac{1}{81}\right)}^{\frac{1}{4}}$

Formula used:

  $a^{\frac{1}{n}}=\sqrt[n]{a}$

  $\sqrt[n]{a^n}=a$

Calculation:

  $\Rightarrow {\left(\frac{1}{81}\right)}^{\frac{1}{4}}$

  $\Rightarrow \sqrt[4]{\frac{1}{81}}$

  $[\because a^{\frac{1}{n}}=\sqrt[n]{a}]$

  $\Rightarrow \sqrt[4]{\frac{1}{3\times 3\times 3\times 3}}$

  $\Rightarrow \sqrt[4]{{\left(\frac{1}{3}\right)}^4}$

  $\Rightarrow \frac{1}{3}$

  $[\because \sqrt[n]{a^n}=a]$

Conclusion:

Therefore, the simplified form of ${\left(\frac{1}{81}\right)}^{\frac{1}{4}}$ is $\frac{1}{3}$ .