Let  n  be an integer such that  n > 1. Then a special name may be used when a quantity  a  is raised to the power  1/n. The expression  a1/n  is called the nth root of  a  that is denoted symbolically as  :

.

In this formula  a  is called the radicand and  n  is said to be the index of the radical.

By the Exponentiation Rules,

,

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So the definition of the nth root can be reformulated like below.

A number  b  is the nth root of  a  if bn = a.

The second and the third roots of  a  have special names, the square root and cube root, respectively.
The index of the square root is omitted from the expression, that is, the square root of  a  is written symbolically as  .

Any number has two square roots whose absolute values are the same.

If  b  is a square root of a positive number  a, that is,  b2 = a, then  –b  is also a square root of a since (–b)2 = b2.
Hence, the square roots of a positive number are two real numbers with the opposite signs. The positive real root is called the principal root.
When an expression refers to a square root of a positive number, this means the principal root of the number. The symbol    is used to represent only the principal root of  a, for instance,  ; both roots of  4  are symbolized by the expression  .
In particular,

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There are no real square roots of a negative number since a real number being squared gives a positive number.
Thus, the square roots of a negative number are two complex numbers.

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