Chapter 6.1, Problem 30OE

To determine

Whether the statement $\sqrt[4]{x^2+1}$ is always a real number, is true or not if $\left(x\right)$ belongs to real.

Answer to Problem 30OE

Yes, the given statement is true.

Explanation of Solution

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

  $\left(\sqrt[4]{x^2+1}\text{ is always a real number}\right)\mathrm{......}\left(1\right)$

Calculation: From (1) as $\left(x\right)$ belongs to real, therefore $\left(x^2\right)$ will also belong to real. Now the argument inside the fourth-root is $\left(x^2+1\right)$ which is also real because $\left(x^2\right)$ is real. And the fourth root of argument $\left(x^2+1\right)$ will also give positive result. Hence it can be said that $\sqrt[4]{x^2+1}$ is always a real number. So the given statement is true.