The Nth Root - Algebra-2

1. Fundamental Concepts

1. Definition

For a real number $a$ and a positive integer $n \ge 2$ :

The nth root of $a$ is a number $x$ such that $x^n = a$ .

Written as $\sqrt[n]{a}$ .

2. Even Roots vs. Odd Roots

Even roots (n is even):

If $a > 0$ , there are two real roots that are opposites of each other.

If $a = 0$ , the only root is $0$ .

If $a < 0$ , there is no real root (result is complex).

Odd roots (n is odd):

Any real number $a$ has exactly one real root (positive, negative, or zero).

3. Principal nth Root

The notation $\sqrt[n]{a}$ denotes the principal root:

If $n$ is even, the principal root is the non-negative real root (or principal complex value).

If $n$ is odd, the principal root is the unique real root.

Roots and Their Properties:
- For even roots of positive numbers, there are two real roots that are opposites of each other. For example, the square roots of 64 are 8 and -8 (i.e., , ).
- For odd roots of negative numbers, there is a unique negative real root. For example, the fifth root of -1 is -1 (i.e., ).
- Negative numbers have no real even roots because the even power of any real number is non-negative.

Easy

Find: $\sqrt[3]{27}$

Solution: Since $3^3 = 27$ , $\sqrt[3]{27} = 3$ .

Medium

Find: The real solutions to $x^4 = 81$

Solution: $$x = \pm \sqrt[4]{81} = \pm \sqrt[4]{3^4} = \pm 3$$

So the solutions are $x = 3$ and $x = -3$ .

Hard

Find: $\sqrt[5]{-243}$

Solution:

$-243 = (-3)^5$ , and $n = 5$ is odd, so:

$$\sqrt[5]{-243} = -3$$

(Note: Odd roots of negative numbers are allowed and yield a single real answer.)

Determine the existence of roots: First, check whether the index is odd or even. If it is even, the radicand must be non-negative to have real roots; if it is odd, any real number has a real root.
Calculate specific roots: Find a number whose nth power equals the radicand; this number is the desired nth root. For common numbers, it can be derived directly through multiplication. For example, to find the fourth root of 81, since and , the fourth roots are 3 and -3.
Handling special values: Remember that any root of 0 is 0 to simplify related calculations.