Chapter 6.2, Problem 44E

To determine

that is the given statement is correct.

Answer to Problem 44E

No

Explanation of Solution

Given:

The statement, “For a real number a and a positive integer n , the value of $\sqrt[n]{a}$ is always positive and the value of $-\sqrt[n]{a}$ is always negative.”

Concept used:

If n is odd then a has one real nth root: $\sqrt[n]{a}=a^{\frac{1}{n}}$

The given statement is incorrect.

For if the real number a is negative and the positive integer n is an odd number, then the value of $\sqrt[n]{a}$ is always negative.

For example, consider $\sqrt[9]{-512}$

Here the index $n=9$ is odd, so $-512$ has only one real ninth root.

Now since ${\left(-2\right)}^9=-512$ , so the real ninth root of $-512$ is $\sqrt[9]{-512}=-2$ which is negative.

And thus, $-\sqrt[9]{-512}=-\left(-2\right)=2$ is positive.

Hence proved, that the given statement is incorrect.