Chapter 6.4, Problem 64HP

To determine

When and why absolute value symbols are needed while taking a $n^{th}$ root.

Answer to Problem 64HP

The absolute value symbols are needed while taking even-roots.

Explanation of Solution

Given information: Consider a number say $\left(x\right)$ .

Calculation: Since the domain of root of odd numbers belongs to all real numbers, while domain of roots of even numbers is defined for the numbers greater than zero. Therefore, while taking roots of even numbers it must be taken in mind that the range of given original expression and domain of result should be same. Mathematically for a number say $\left(x\right)$ ,

  $\left(\begin{array}{c}\sqrt[4]{\left(x^4\right)}=x\mathrm{......}\left(\text{is not true}\right)\\ \text{while}\sqrt[4]{\left(-1^4\right)}=-\mathrm{1......}\left(\text{when }x=-1\right)\\ 1=-\mathrm{1......}\left(\text{contradiction}\right)\end{array}\right)\mathrm{......}\left(1\right)$

But, with absolute brackets (1) holds true as shown

  $\left|1\right|=\left|-1\right|\mathrm{......}\left(\text{no}-\text{contradiction}\right)$

Hence, the valid statement is $\sqrt[4]{\left(x^4\right)}=\left|x\right|$.