nth Roots and Radicals •

a

is the nth root of

b

if and only if

a n



b

• Example 1: 2 - 3 is the third root is the fifth root of 8 , since 2 3 of - 243 , since



8   5   243

• As we saw with square roots, when

n

is even, there can be two possible

n

th roots.

• Example 2: 3 is the fourth root of 81 , since 3 4  81 - 3 is also the fourth root of 81 , since   4



81

• 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   4

 

16 Note that neither 2 nor – 2 will work.

  4



16   4



16

• A common way to express

n

th roots is using radical notation .

n b b

is called the radicand

n

is called the index is called a radical symbol

n b

is called a radical

• Example 4 5 32 32 is the radicand 5 is the index

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

36



2 36

• Since 2 4  16 and   4



16 both 2 and -2 are fourth roots of 16 .

• Expressed in radical notation … 4 16



2 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

n

th root of

b

, which is the non-negative root .

• Example 6 Consider the radical: 4 81 There are two fourth roots of 81 , - 3 and 3 .

The principal fourth root is the non-negative root, or 3 . Therefore, 4 81



3

• Example 7 Consider the radical: 3



8 With an odd index, there is only one third root of – 8 . Therefore, 3 2

• Example 8 Evaluation 4 81



3 5 32



2 3 64



4 Reasoning 3 4  81 2 5



32 4 3



64

• Example 9 Evaluation 3 2 5 1 Reasoning   3   8   5

 

1

• 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 9 3 27 Square root of 9 Cube root of 27