We have seen that raising to a power is the inverse operation for extracting roots, so that
For example,
What if the power and root operations occur in the opposite order? Is it always true that
First, consider the case where the index is an odd number. For example,
Because every real number has exactly one th root if is odd, we see that,
However, if is even, then is positive, regardless of whether itself is positive or negative, and hence is positive also. For example, if then
In this case, does not equal because is negative but is positive. We must be careful when taking even roots of powers. We have the following special relationship for even roots.
In particular, note that it is not always true that
unless we know that
Otherwise, we can only assume that