Chapter 6, Problem 1RE

To determine

Whether the answer is true or false.

Answer to Problem 1RE

Solution:

True

Explanation of Solution

Every positive real number a has a single positive n^throot, called the principal n^throot which is written as $\sqrt[n]{a}$.

An n^throot of a number a, where n is usually assumed to be a positive integer, is a number b which when raised to the power n yields b , i.e., if $b^n=a$, then $b={\left(a\right)}^{\frac{1}{n}}$ is the n^throot of a, here n is the degree of the root.

Any non-zero number complex number has n distinct complex roots of degree n with real roots also.

If n is even and a > 0 where $a\in R$, then one of its n^throot is positive, one is negative and the rest are n a considered as complex number has n different “complex roots of degree n, including those with zero imaginary part, i.e., any real roots.

If n is even and a > 0 where $a\in R$, one of its n^throot is positive, one is negative and the rest are complex root but not real.

If n is even and a < 0 where $a\in R$, none of its n^throot is real.

If n is odd and a > 0 where $a\in R$, one of its n^throot is real and has the same sign as a, while the other (n - 1) roots are not real.

Finally if a is not real, then none of its n^throot is real.

Given:

The principal n^throot of an odd-indexed root is always positive.

Conclusion:

The given statement is true.