Principal nth root

Definition

The principal nth root of a number $a$ is the unique real number $b$ such that $b^n = a$, where $n$ is a positive integer. When $n$ is even, the principal nth root is non-negative.

5 Must Know Facts For Your Next Test

1. $\sqrt[n]{a}$ represents the principal nth root of $a$.
2. For any real number $a \geq 0$ and even integer $n$, the principal nth root is always non-negative.
3. When $n$ is odd, every real number $a$ has exactly one real nth root which could be negative or positive.
4. The principal square root (when $n = 2$) of a non-negative number is always positive or zero.
5. The notation for the principal nth root can also be expressed using rational exponents as $a^{1/n}$.

Review Questions

• What does the symbol $\sqrt[3]{-8}$ represent?
• If $x = \sqrt[4]{16}$, what is the value of x?
• How do you express the principal fifth root of 32 using rational exponents?

Related terms

Radical: An expression that includes a root symbol ($\sqrt{}$) indicating a need to find some specific factor of a given quantity.

Rational Exponent: An exponent expressed as a fraction, where the numerator indicates power and the denominator indicates root, such as in $a^{m/n}$.

Square Root: The value that, when multiplied by itself, gives the original number. It is denoted by the radical symbol with an index of 2.