Chapter 6.1, Problem 25OE

To determine

Whether $\sqrt[3]{x}$ is always a real number or not if $\left(x\right)$ belongs to real.

Answer to Problem 25OE

Yes, the given statement is true.

Explanation of Solution

Given information: The given statement for variable $\left(x\right)$ which belongs to real is,

  $\left(\sqrt[3]{x}\text{ is always a real number}\right)\mathrm{......}\left(1\right)$

Calculation: From (1),

  $\left(\begin{array}{c}\sqrt[3]{x}=\text{positive}\mathrm{......}\left(\text{For }x>0\right)\\ =\text{negative}\mathrm{......}\left(\text{For }x<0\right)\\ =\mathrm{0......}\left(\text{For }x=0\right)\end{array}\right)\mathrm{......}\left(2\right)$

From (2) the cube-root of $\left(x\right)$ belongs to real whether $\left(x\right)$ is positive, negative or zero. Hence it can be said that $\sqrt[3]{x}$ is always a real number. So the given statement is true.