Beginning Algebra, 11th Edition

SECTION 8.3 Adding and Subtracting Radicals 513

OBJECTIVES

Adding and Subtracting Radicals

8.3

1 Add and subtract

radicals.

2 Simplify radical

sums and

differences.

3 Simplify more

complicated radical

expressions.

OBJECTIVE 1 Add and subtract radicals.We add or subtract radicals as shown.

= 1423 Add. Subtract. =- 5211

= 18 + 6223 = 12 - 72211

823 + 623 2211 - 7211

(b)

=- 2210

= 15 - 72210

5210 - 7210

(c)

= 327

= 11 + 2227

= 127 + 227

27 + 227 (d)

= 225

= 11 + 1225

= 125 + 125

25 + 25

OBJECTIVE 2 Simplify radical sums and differences.

Simplifying Radicals to Add or Subtract

Add or subtract, as indicated.

(a)

Factor; 4 is a perfect square.

Product rule

Distributive property

Add.

(b)

Factor; 9 is a perfect square.

Product rule

= 322 - 323 29 = 3

= 29 # 22 - 29 # 23

= 29 # 2 - 29 # 3

218 - 227

= 522

= 13 + 2222

= 322 + 222 24 = 2

= 322 + 24 # 22

= 322 + 24 # 2

322 + 28

EXAMPLE 2

These are unlike

radicals. They cannot

be combined.

NOW TRY
EXERCISE 1
Add or subtract, as indicated.

(a)

(b)

(c) 25 + 214

2211 - 6211

423 + 23

NOW TRY ANSWERS

1. (a) (b)
   (c)It cannot be added by the
   distributive property.

523 - 4211

Distributive

property

Distributive

property

Only like radicals— those that are multiples of the same root of the same number—

can be combined in this way. By contrast, examples of unlike radicalsare

Radicands are different.

as well as Indexes are different.

Adding and Subtracting Like Radicals

Add or subtract, as indicated.

(a)

= 826

= 13 + 5226

326 + 526

EXAMPLE 1

223 and 2 233.

225 and 2 23 ,

(e) cannot be

added by the distributive

property. They are unlike

radicals.

NOW TRY

23 + 27

We are factoring

out here. 26