§ 7.4

Adding, Subtracting, and Dividing Radical Expressions

Blitzer, Intermediate Algebra, 5e – Slide #2 Section 7.4

Combining Radicals

Apples to apples, oranges to oranges,… you can only add “like” things….

Two or more radical expressions that have the same indices and the same radicands are called like radicals.

Like radicals can be combined under addition in exactly the same way that we combined like terms under addition. Examples of this process follow.

2 elephants + 3 elephants = 5 elephantsbut

5 tigers + 3 gorillas = ???

Blitzer, Intermediate Algebra, 5e – Slide #3 Section 7.4

Combining Radicals

EXAMPLEEXAMPLE

Simplify (add or subtract) by combining like radical terms: .57276(b)7479(a) 3333 xx

SOLUTIONSOLUTION

Apply the distributive property.

Simplify.

33 7479(a) 3 749

3 7533 57276(b) xx

33 57276 xx

Apply the distributive property.

Group like terms.

351726 x

Simplify.3478 x

Blitzer, Intermediate Algebra, 5e – Slide #4 Section 7.4

Combining Radicals

EXAMPLEEXAMPLE

Simplify by combining like radical terms, if possible:

.54(b)3124(a) 3 23 24 xyxyxxx

SOLUTIONSOLUTION

Factor the radicands using the greatest perfect square factors.

Apply the distributive property.

Take the square root of each factor.

3124(a) xx 334 xx

334 xx

332 xx 24

312 x

Blitzer, Intermediate Algebra, 5e – Slide #5 Section 7.4

Combining Radicals

3 23 24 54(b) xyxyx Factor the radicands using the greatest perfect cube factors.

Simplify.

Take the cube root of each factor.

33 x

CONTINUECONTINUEDD

3 23 23 54 xyxxyx

3 23 23 3 54 xyxxyx

Apply the distributive property.

3 23 2 54 xyxxyx xx 3 3

3 254 xyxx

Simplify.3 29 xyx

Blitzer, Intermediate Algebra, 5e – Slide #6 Section 7.4

Simplifying Radicals

The Quotient Rule for RadicalsIf and are real numbers and , then

The nth root of a quotient is the quotient of the nth roots.

n a n b 0b

.n

n

n

b

a

b

a

Blitzer, Intermediate Algebra, 5e – Slide #7 Section 7.4

Simplifying Radicals

EXAMPLEEXAMPLE

Simplify using the quotient rule: .13

(b)81

50(a) 4

12

7

8

3

x

y

y

x

SOLUTIONSOLUTION

44

2

4

2

8

3

8

3

9

25

9

225

9

225

81

50

81

50(a)

y

xx

y

xx

y

xx

y

x

y

x

3

4 3

3

4 34 4

3

4 34

4 12

4 7

412

7 1313131313(b)

x

yy

x

yy

x

yy

x

y

x

y

Blitzer, Intermediate Algebra, 5e – Slide #8 Section 7.4

Combining Radicals

Dividing Radical ExpressionsIf and are real numbers and , then

To divide two radical expressions with the same index, divide the radicands and retain the common index.

n a n b 0b

.nn

n

b

a

b

a

Blitzer, Intermediate Algebra, 5e – Slide #9 Section 7.4

Combining Radicals

EXAMPLEEXAMPLE

Divide and, if possible, simplify: .2

250(b)

3

54(a)

3 3

3 35

24

117

x

yx

ba

ba

SOLUTIONSOLUTION

In each part of this problem, the indices in the numerator and the denominator are the same. Perform each division by dividing the radicands and retaining the common index.

24

117

24

117

3

54

3

54(a)

ba

ba

ba

ba

2114718 ba

Divide the radicands and retain the common index.

Divide factors in the radicand. Subtract exponents on common bases.

Blitzer, Intermediate Algebra, 5e – Slide #10 Section 7.4

Combining Radicals

33

35

3 3

3 35

2

250

2

250(b)

x

yx

x

yx

131118 ba

CONTINUECONTINUEDD Simplify.

abba 29 1210Factor using the greatest perfect square factor.

abba 29 1210 Factor into two radicals.

Simplify.abba 23 65

3 335125 yx

Divide the radicands and retain the common index.

Divide factors in the radicand. Subtract exponents on common bases.

Blitzer, Intermediate Algebra, 5e – Slide #11 Section 7.4

Combining Radicals

CONTINUECONTINUEDD Simplify.

Factor using the greatest perfect square factor.

Factor into two radicals.

Simplify.

3 32125 yx

3 23125 xy

3 23 3125 xy 3 25 xy

Blitzer, Intermediate Algebra, 5e – Slide #12 Section 7.4

Combining Radicals

Important to remember:

Like radicals have the same indices and radicands. Like radicals can be added or subtracted using the distributive property.

In some cases, you cannot see that radicals are “like” until you simplify them. When attempting to combine radicals, you should simplify the radicals first. Then you may see that youhave like radicals that can be combined.

37

3532

7512

Are we like? You don’t look like me.Yep. I’m 2 square roots of 3 and you are 5 square roots of 3. We have the same indices and radicands. We’re like!Let’s see…2 of them + 5 of them = 7 of them