Chapter 6.1, Problem 28OE

To determine

Whether the statement $\sqrt[3]{x+1}$ is always a real number, is true or not if $\left(x\right)$ belongs to real.

Answer to Problem 28OE

Yes, the given statement is true.

Explanation of Solution

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

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

Calculation: From (1),

  $\left(\begin{array}{c}\sqrt[3]{x+1}=\text{real}\mathrm{......}\left(\text{For }x\ge -1\right)\\ =\text{real}\mathrm{......}\left(\text{For }x<-1\right)\end{array}\right)\mathrm{......}\left(2\right)$

From (2) the statement $\sqrt[3]{x+1}$ gives real values for all $\left(x\right)$ belongs to real. Hence it can be said that $\sqrt[3]{x+1}$ is always a real number. So the given statement is true.