Chapter 7, Problem 20MCQ

To determine

To Simplify: The expression ${\left(\frac{16}{81}\right)}^{\frac{1}{4}}$

Answer to Problem 20MCQ

  ${\left(\frac{16}{81}\right)}^{\frac{1}{4}}=\frac{2}{3}$

Explanation of Solution

Given information:

The expression ${\left(\frac{16}{81}\right)}^{\frac{1}{4}}$

Concept and Formula Used:

Rule of exponent.

  $\sqrt[n]{a}=a^{\frac{1}{n}},{\left(ab\right)}^n=a^n{b}^n$

BODMAS rule

Calculation:

The given expression is

  $\begin{array}{l}{\left(\frac{16}{81}\right)}^{\frac{1}{4}}={\left(\frac{2\times 2\times 2\times 2}{3\times 3\times 3\times 3}\right)}^{\frac{1}{4}}\\ {\left(\frac{16}{81}\right)}^{\frac{1}{4}}={\left[{\left(\frac{2}{3}\right)}^4\right]}^{\frac{1}{4}}\\ \text{Using }{\left(a^m\right)}^n=a^{m\times n}\\ {\left(\frac{16}{81}\right)}^{\frac{1}{4}}={\left[\left(\frac{2}{3}\right)\right]}^{4\times \frac{1}{4}}\\ {\left(\frac{16}{81}\right)}^{\frac{1}{4}}={\left[\frac{2}{3}\right]}^1\\ {\left(\frac{16}{81}\right)}^{\frac{1}{4}}=\frac{2}{3}\end{array}$

Conclusion:

  ${\left(\frac{16}{81}\right)}^{\frac{1}{4}}=\frac{2}{3}$