Beginning Algebra, 11th Edition

8.1 Evaluating Roots

Ifais a positive real number, then

is the positive square root of a;

is the negative square root of a;

Ifais a negative real number, then is not a real
number.

Ifais a positive rational number, then is rational if
ais a perfect square and is irrational if ais not a
perfect square.

2 a

2 a

2 a

 2 a 20 =0.

2 a

is not a real number.

are rational. 221 are irrational.

B

2

3

216 ,

B

4

9

,

2 - 25

• 281 =- 9

249 = 7

QUICK REVIEW

CONCEPTS EXAMPLES

CHAPTER 8 Summary 545

Distance Formula
The distance between and is

d 21 x 2 x 122  1 y 2 y 122.

1 x 1 ,y 12 1 x 2 ,y 22

The distance between and is

= 210.

= 21 + 9

= 21 - 122 + 32

21 - 1 - 022 + 31 - 1 - 2242

1 0,- 22 1 - 1, 1 2

Each real number has exactly one real cube root. 2327 = 3 23 - 8 =- 2

8.2 Multiplying, Dividing, and Simplifying

Radicals

Product Rule for Radicals
For nonnegative real numbers aandb,

and

Quotient Rule for Radicals
Ifaandbare nonnegative real numbers and then

and

If all indicated roots are real, then

and

2 na

2 nb

n

A

a

b

2 na# 2 nb 2 nab 1 b 02.

A

a

b



2 a

2 b

.

2 a

2 b



A

a

b

bZ0,

2 a# 2 b 2 ab 2 a#b 2 a# 2 b.

2412

244

=

B

4

12

4

235 # 233 = 2315 = 243

B

25

64

=

225

264

=

5

8

28

22

=

B

8

2

= 24 = 2

248 = 216 # 3 = 216 # 23 = 423

25 # 27 = 235

8.3 Adding and Subtracting Radicals

Add and subtract like radicals by using the distributive
property. Only like radicals can be combined in this way.
= 625 =- 222

= 12 + 4225 = 222 - 422

225 + 425 28 - 232

8.4 Rationalizing the Denominator

The denominator of a radical can be rationalized by
multiplying both the numerator and denominator by
a number that will eliminate the radical from the
denominator. B

3

5

6

=

235 # 2362

236 # 2362

=

23180

6

2

23

=

2 # 23

23 # 23

=

223

3

(continued)