Consider the polynomial equation $x^{n} = b .$ The Fundamental Theorem of Algebra tells us that this polynomial equation has $n$ roots, which can be either real or imaginary. When restricting ourselves to just the real ones, we refer to a number $a,$ such that $a^{n} = b,$ as the real $n th$ root of $b .$ There are two different cases.

Solving $a^{n} = b$ When $n$ Is Odd

If $n$ is odd, there is just one real $n th$ root of $b,$ which we denote as $n b .$

Equation	Real Solutions	$n th$ Root
$x^{3} = 8$	$x = 2$	$38 = 2$
$x^{3} = - 27$	$x = - 3$	$3 - 27 = - 3$

Solving $a^{n} = b$ When $n$ Is Even

If $n$ is even, there are two possibilities, depending on if $b$ is positive or negative. If $b$ is positive there are two possible real $n th$ roots. The positive root is know as the principal root.

Equation	Real Solutions	$n th$ Root	Root Type
$x^{4} = 16$	$x = 2$	$416 = 2$	Principal Root
$x^{4} = 16$	$x = - 2$	$- 416 = - 2$	Negative Root

On the other hand, if $b$ is negative, there are no real solutions. All the $n th$ roots are imaginary in that case.

Conclusions

A solution to the equation $x^{n} = b$ is called the $n th$ root of $b .$

We use the term real $n th$ root to differ between real and complex possible roots.
We use the term principal root to refer to the positive real root in cases when we can have a positive and a negative real root.

Solving an=b When n Is Odd

Solving an=b When n Is Even

Conclusions

Solving $a^{n} = b$ When $n$ Is Odd

Solving $a^{n} = b$ When $n$ Is Even

Exercise