# Radical

A radical expression, also referred to as an n^{th} root, or simply radical, is an expression that involves a root. Radicals are expressed using a radicand (similar to a dividend), a radical symbol, and an index, which is typically denoted as "n." The most common radicals we see are the square root and the cubed root. The square root is so commonly used that by convention, a radical written without an index is assumed to be a square root.

The above figure, as a whole, makes up a radical. It is read as "the n^{th} root of (x + 2)." If n were 3, it would be the cubed root; if it were 2, it would be the square root. The n^{th} root of a radicand is equal to the value, that raised to the n^{th} power, would equal the radicand. Note that radicals and exponents are closely related, and a radical can be written as the radicand raised to the power of

Examples

1. Find :

This is a simple example for the purpose of demonstrating what a radicand is. As mentioned, a radicand written without an index is assumed to be a square root. We could also have written the above problem as:

When evaluating the square root, we are looking for a value, x, that raised to the power of 2, equals the radicand. In this case, 2^{2} = 4, so 2 is a square root of 4.

2. Find :

The cubed root, like the square root, and n^{th} root, is found in the same way. The problem above can be read as: what value, raised to the power of 3, equals 8? The answer is 2 since:

2^{3} = 2 × 2 × 2 = 8

## Properties of radicals

There are many properties of radicals and exponents that can be helpful for simplifying expressions or solving equations. Below are some of them.

(1) |

(2) |

Note: While the properties above might seem like they could apply to addition and subtraction, it is important to note that they do not apply to addition and subtraction. We cannot separate addition and subtraction under a radical the same way we do multiplication and division.

(3) If then |

(z must be ≥ 0 for even n) |

(4) |

(if x ≥ 0 or if n is odd) |

(5) |

(when x < 0 and n is even) |

(6) |

There are other properties or ways we can manipulate radicals and exponents, but these are some of the more common properties that can be helpful for solving equations or simplifying expressions.