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Chapter 10

Radicals and Rational Radicals and Rational ExponentsExponents

10.1 Finding Roots

10.2 Rational Exponents

10.3 Simplifying Expressions Containing Square Roots

10.4 Simplifying Expressions Containing Higher Roots

10.5 Adding, Subtracting, and Multiplying Radicals

10.6 Dividing RadicalsPutting it All Together

10.7 Solving Radical Equations

10.8 Complex Numbers

1010 Radicals and Rational Radicals and Rational ExponentsExponents

Simplifying Expressions Containing Square RootsSimplifying Expressions Containing Square Roots10.310.3

Multiply Square Roots

.164product thebegin with sLet'

There are two ways that the product can be found. Let’s take a look at both ways.

First method is to find the square roots of each and then multiply the results.

164 42 8

Second method is to multiply the radicands and then find the square root.

164 164 864 Notice that both will obtain the same results. This leads us to the product rule forMultiplying expressions containing square roots.

Example 1

mba 5)37)

Multiply.

Solution

37)a 37 21

mb 5) m5 m5

BeCareful

We can multiply radicals this way only if the indices are the same. Wewill see later how to multiply radicals with different indices such as 321 t

Simplify the Square Root of a Whole Number

How do we know when a square root is simplified?

To simplify expressions containing square roots, we reverse the process of multiplying. That is we use the product rule that says where a andb is a perfect square.

baba

Example 2

15)500)75)45) dcba

Simplify completely.

Solution

a) The radical is not in simplest form since 45 contains a factor (other than 1) That is a perfect square. Think of two numbers that multiply to 45 so that at leastOne of the numbers is a perfect square:

45

5945

Notice, however that ,but neither 3 or 15 is a perfect square. So we need to use 9 and 5.

15345

45 59

59

53

9 is a perfect square.

Product Rule.

39

squares.perfect

are that factorsany havenot does 5 because simplified completely is 53

Examples b through d continued on next page…

Example 2

15)500)75)45) dcba

Simplify completely.

Solution

b) The radical is not in simplest form since 75 contains a factor (other than 1) Think of two numbers that multiply to 75 so that at least one of the numbers is a perfect square:

75

32575

Notice, 25 is a perfect square and 3 is not a perfect square.

75 325

325

35

25 is a perfect square.

Product Rule.

525

squares.perfect

are that factorsany havenot does 5 because simplified completely is 35

Examples c continued on next page…

Example 2

15)500)75)45) dcba

Simplify completely.

Solution

c) The radical is not in simplest form since 500 contains a factor (other than 1). Think of two numbers that multiply to 500 so that at least one of the numbers is perfect square:

500

5100500

Notice, 100 is a perfect square and 5 is not a perfect square.

500 5100

5100

510

25 is a perfect square.

Product Rule.

10100

squares.perfect

are that factorsany havenot does 5 because simplified completely is 510

Examples c continued on next page…

d) Does 48 have a factor that is a perfect square? 31648 Yes,

asit rewrite ,48simpify To

48 316

316 34

16 is a perfect square.

Product Rule.

416

34However, notice that .12448

48 124124

122122

Notice that is not in simplest form. We must continue to simplify.

12

48 122342

342 322

34

Example 2-Continued

48)15)75)45) dcbaSimplify completely.

Solution

Use the Quotient Rule for Square Roots

roots. square containing sexpression dividingfor rulequotient the tous leads This

.249

36

9

36 that truealso isIt .2

3

6

9

36say can We.

9

36simplify sLet'

Example 3

49

9)a

Simplify completely.

Solution

49

9

49

9

7

3

Quotient Rule

Since 9 and 49 are perfect squares, find the square root of each separately.

749 and 39

Example 4

3

300)a

Simplify completely.

Solution

3

300100

10

Simplify

10

120)b

36

5)c

a) Neither 300 nor 3 are perfect squares, so you want to simplify to get 100, which is a perfect square. 3

300

3

300

b) Neither 120 nor 10 are perfect squares. There are two methods that you can use to simplify One is to apply the quotient rule to obtain a fraction under one radical and then simplify and the second is to apply the product rule to rewrite each radical and then simplify the fraction.

10

120

10

120

12 34 32

Quotient rule

Square root of 4 is 2.

10

120

25

304

25

652

25

2352

32

Product rule

Divide out common Factors.

Method 2Method 1

Example 4-Continued

3

300)a

Simplify completely.

10

120)b

36

5)c

Solution

c) The fraction does not reduce. However, 36 is a perfect square. Begin by applying the quotient rule.

36

5

36

5

36

5

6

5

Quotient rule

636

Multiplying is the same as dividing 6 by 2. We can generalize this result withThe following statement.

Simplify Square Root Expressions Containing Variables with Even Exponents

A square root is not simplified if it contains any factors that are perfect squares.This means that a square rot containing variables is simplified if the power on eachvariable is less than 2. For example, is not in simplified form. If r represents anonnegative real number, then we can use rational exponents to simplify .

2/16

6r6r

6r 32/62/162/16 rrrr

We can combine this property with the product and quotient rules to simplify radicalexpressions.

Example 5

2) ba

Simplify completely.

Solution

4100) ab 824) pc

2) ba 2

12

b 2/2b 1b b

4100) ab 4100 a 2/410 a 210a

824) pc 824 p

2/864 p

462 p

62 4p

6

27)g

d

6

27)g

d6

27

g

2/6

39

g

2/6

39

g

3

33

g

Product rule

4 is a perfect square.

Simplify.

Rewrite using the commutative property.

Begin by using the quotient rule.

Simplify Square Root Expressions Containing Variables with Odd Exponents

How do we simplify an expression containing a square root if the power under theSquare root is odd? We can use this product rule for radicals and fractional exponents to help us understand how to simplify such expressions.

Example 6

9) ba

Simplify completely.

13) ab

9) ba

428

SolutionTo simplify the radicals , write the variable as the product of two factors so that the exponent of one of the factors is the largest numbers less than 9 that is divisible by 2 (the index of the radical).

18 bb

bb 8

bb 2/8

bb4

8 is the largest number lessthan 9 that is divisible by 2.

Product Rule

Use a fractional exponentto simplify.

13) ab

6212

112 aa

aa 12

aa 2/12

aa6

12 is the largest number less than 9 that is divisible by 2.

Product Rule

Use a fractional exponent to simplify.

If you notice in the previous examples, we always divided by 2. Let’s look at theprevious examples to see if we can used division to help us simplify radicals.

9b 18 bb bb4

1

8

492

Quotient

Remainder

Index of radical

13a 112 aa aa6

1

12

6132

Quotient

Remainder

Index of radical

Example 7

5) ka

Simplify completely.

349) vb 1932) mcSolution

:divide ,simplify To) 5ka25241kkkkk 2125

349) vb 349 v

vv7

vv7

Product Rule

1. ofremainder a and 2 ofquotient a gives 23

1932) mc 1932 m

mm9216

mm924

Product Rule

1. ofremainder a and 9 ofquotient a gives 219

mm 24 9

mm 24 9

416

Use commutative property to rewrite expression

Use the product rule to write the expression with one radical.

Example 8

9718) nma

Simplify completely.

12

93)

g

fb

9718) nma

Solution

9718 nm 141329 nnmm

Use the Product Rule for each Radical.

141323 nnmm 39

nmnm 23 43

mnnm 23 43

Use commutative property to rewrite expression

Use the product rule to write the expression with one radical.

12

93)

g

fb

12

93

g

f

12

93

g

f

6

143

g

ff

6

14 3

g

ff

6

4 3

g

ff

Quotient Rule

Product Rule

Use commutative property to rewrite expression

Use the product rule to write the expression with one radical.

Simplify More Square Root Expressions Containing VariablesExample 9

bba 28)

Simplify completely.

45 63) xyyxb 3

7

2

36)

r

rc

Solutionbba 28) bb 28

216b

216 b

b4

b4

Product Rule

Product Rule

45 63) xyyxb 5618 yx5618 yx

5629 yx 5629 yx

yyx 2323 yyx 23 23 yyx 23 23

Multiply the radicands together to obtain one radical.

Product Rule

Product Rule

Evaluate Commutative property

Example 9-Continued

bba 28)

Simplify completely.

45 63) xyyxb 3

7

2

36)

r

rc

Solution

3

7

2

36)

r

rc

3

7

2

36

r

r

418r

429 r

223 r

23 2r

Use Quotient Rule.

Product Rule.

Commutative property