Beginning Algebra, 11th Edition

Finding Cube Roots

Find each cube root.

(a) What number can be cubed to give 8? Because

(b) , because

(c) because. NOW TRY

Notice in Example 9(b)that we can find the cube root of a negative number.

(Contrast this with the square root of a negative number, which is not real.) In fact,

the cube root of a positive number is positive, and the cube root of a negative number

is negative. There is only one real number cube root for each real number.

When a radical has an even index(square root, fourth root, and so on), the radi-

cand must be nonnegativeto yield a real number root. Also, for

and so on are positive (principal) roots.

and so on are negative roots.

Finding Other Roots

Find each root.

(a) because 2 is positive and

(b) From part (a), so the negative root is

(c) For a fourth root to be a real number, the radicand must be non-

negative. There is no real number that equals

(d)

First find. Because 2 is the number whose fifth power is 32,.

Since , it follows that

(e) 25 - 32 =- 2 , because 1 - 225 =-32. NOW TRY

- 2532 =-2.

2532 = 2

2532 2532 = 2

- 2532

24 - 16.

24 - 16

- 2416 2416 =2, - 2416 = -2.

2416 =2, 24 =16.

EXAMPLE 10

 2 a, 24 a, 26 a,

2 a, 24 a, 26 a,

a 7 0,

23216 =6, 63 = 216

23 - 8 =- 2 1 - 223 =-8.

238 23 =8, 238 =2.

EXAMPLE 9

500 CHAPTER 8 Roots and Radicals

NOW TRY
EXERCISE 9
Find each cube root.

(a) (b)

(c) 2327

23343 23 - 1000

NOW TRY
EXERCISE 10
Find each root.

(a) (b)

(c) 253125 (d) 25 - 3125

24625 24 - 625

NOW TRY ANSWERS

9. (a) 7 (b) (c) 3


10. (a) 5 (b)not a real number
    (c) 5 (d)- 5

• 10

Complete solution available
on the Video Resources on DVD

8.1 EXERCISES

Concept Check Decide whether each statement is trueor false.If false, tell why.

1.Every positive number has two real

square roots.

2.A negative number has negative real

square roots.

3.Every nonnegative number has two real

square roots.

4.The positive square root of a positive

number is its principal square root.

5.The cube root of every nonzero real num-

ber has the same sign as the number itself.

6.Every positive number has three real

cube roots.

Find all square roots of each number. See Example 1.

7. 9 8. 16 9. 64 10. 100 11. 169


8. 225 13. 14. 15. 900 16. 1600
   81
   400

25

196

http://www.ebook777.com

http://www.ebook777.com