nth Roots and Radicals Example 1: a is the nth root of b if and only if 2 is the third root of 8, since - 3 is the fifth root of - 243, since

Example 2: As we saw with square roots, when n is even, there can be two possible n th roots. 3 is the fourth root of 81, since - 3 is also the fourth root of 81, since

Sometimes there are no n th roots of a number. Example 3: There are no fourth roots of – 16 since there are no numbers such that Note that neither 2 nor – 2 will work.

A common way to express nth roots is using radical notation. b is called the radicand n is called the index is called a radical symbol is called a radical

Example 4 32 is the radicand 5 is the index

Example 5 36 is the radicand Since there is no number written in the index position, it is assumed to be a 2, or square root.

both 2 and -2 are fourth roots of 16. Since and Expressed in radical notation … Read this as the fourth root of 16 is 2. Note that we use only the positive 2 and not the negative 2.

As with square roots, when n is even and radical notation is used, the result is the principal nth root of b, which is the non-negative root. Example 6 Consider the radical: There are two fourth roots of 81, - 3 and 3. The principal fourth root is the non-negative root, or 3. Therefore,

Example 7 Consider the radical: With an odd index, there is only one third root of – 8. Therefore,

Example 8 Evaluation Reasoning

Example 9 Evaluation Reasoning

As shown in previous slide shows, there are special ways to describe radicals when the index is either a 2 or a 3. Index = 2: Square Root Index = 3: Cube Root Example 10 Square root of 9Cube root of 27