Easy Algebra Step-by-Step

38 Easy Algebra Step-by-Step

In general, if xxx xa

nx

xxx⋅⋅x

ft f

, where n is a natural number, x is called an

nth root of a. The principal nth root of a is denoted na. The expression na

is called a radical, a is called the radicand, n is called the index and indi-

cates which root is desired. If no index is written, it is understood to be 2

and the radical expression indicates the principal square root of the radi-

cand. As a rule, a positive real number has exactly one real positive nth

root whether n is even or odd, and every real number has exactly one real

nth root when n is odd. Negative numbers do not have real nth roots when

n is even. Finally, the nth root of 0 is 0 whether n is even or odd: n 00

(a lway s).

Problem Find the indicated root, if possible.

a.^481

b. −

1

32

5

c.^30. 125

d.^6 − 1

e.^7 − 1

f.^500

Solution

a.^481

Step 1. Find the principal fourth root of 81.

33333333 = 81 , so^481 = 3.

b. −^1

32

5

Step 1. Find the principal fi fth root of −^1

32

.

−⋅−⋅−⋅−⋅−=−

1

2

1

2

1

2

1

2

1

2

1

32

, so −=−

1

32

1

2

(^5).