Principal nth root

Definition

The principal nth root of a number is the non-negative root when n is even and the real root when n is odd. It is denoted as $\sqrt[n]{a}$ where $a$ is the radicand and $n$ is the index.

5 Must Know Facts For Your Next Test

1. The principal square root (n=2) of a number \(a\) is written as \(\sqrt{a}\).
2. For even values of \(n\), there are two roots: one positive (principal) and one negative.
3. For odd values of \(n\), there is only one real nth root, which can be positive or negative depending on \(a\).
4. If \(a \geq 0\) and \(n\) is even, the principal nth root will always be non-negative.
5. The principal nth root function, \(f(x) = \sqrt[n]{x}\), has different properties for different values of \(n\); it’s continuous and defined for all real numbers when \(n\) is odd.

Review Questions

• What distinguishes the principal nth root from other roots when n is even?
• How does the value of n affect the number and type of roots?
• Explain why the principal nth root function behaves differently for even and odd values of n.

Related terms

Radicand: The number under the radical sign in a radical expression.

Index: The small number outside and to the left of the radical sign indicating which root to take.

\text{Rational Exponent}: \text{An exponent that can be expressed as a fraction } \frac{m}{n}, \text{ indicating an nth root raised to an mth power}.