Chapter A.1, Problem 85E

Solution

To explain why $\sqrt[n]{a}$ and a real nth root of a need not have the same value.

Given:

Why $\sqrt[n]{a}$ and a real nth root of a need not have the same value.

Calculation:

The value of $\sqrt[n]{a}$ depends on the index n and the integrand a . If the index is an odd integer, there will always be at least one real root for the real number a . But if the index n is an even integer, then there will be real root only when the integrand a is greater than or equal to 0.

As the even root of any negative real number always gives the imaginary roots.