Chapter 6.2, Problem 29E

To determine

To verify: the error in rewriting the expression ${\left(\sqrt[3]{2}\right)}^4$ in rational exponent form as:

  ${\left(\sqrt[3]{2}\right)}^4=2^{\frac{3}{4}}$

Answer to Problem 29E

The error is that the denominator and numerator (of the fraction as power) are interchanged.

  ${\left(\sqrt[3]{2}\right)}^4=2^{\frac{4}{3}}$

Explanation of Solution

Formula Used:

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

Calculation:

The error is that the denominator and numerator (of the fraction as power) are interchanged.

Correct way to rewrite the expression ${\left(\sqrt[3]{2}\right)}^4$ in rational exponent form is:

  $\begin{array}{c}{\left(\sqrt[3]{2}\right)}^4={\left[{\left(2\right)}^{\frac{1}{3}}\right]}^4\\ =2^{\frac{4}{3}}\end{array}$