1.5.a radicals

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MATH 17 - COLLEGE ALGEBRA AND TRIGONOMETRY

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RATIONAL EXPONENTS & RADICALS

2

OBJECTIVES

Upon completion, you should be able to

Simplify radicals; and

Perform addition, subtraction, multiplication and division of radicals.

RATIONAL EXPONENTS & RADICALS

RADICALS

A radical (or irrational expression) is an

algebraic expression involving rational

exponents. It is of the form:

RATIONAL EXPONENTS & RADICALS

n

mn m aa

3

Examples

Write as a radical:

RATIONAL EXPONENTS & RADICALS

4

3

2

1

b2a .1 4 32 ba

2

1

2

1

3

1

3

1

x

x .2

y

y

yx

yx

33

4

Examples

Write the following radicals in exponential

form:

RATIONAL EXPONENTS & RADICALS

3 2zxy .1 3

12zxy 3

1

3

2

3

1

zyx

6 5

4 33

.2yx

yx

6

5

2

1

4

3

3

1

yx

yx

5

Properties of Radicals

Theorem. If a and b are nonnegative real

numbers and n is even, then:

RATIONAL EXPONENTS & RADICALS

1 1 1

1) n n nn n nab ab a b a b 1 1

12) , 0n n

n

n

nn

a a a ab

b b b b

3) nn a a11

Properties of Radicals

Remarks.

1. If n is odd, properties one and two are

true.

2. For any real number a, then

provided n is even.

3. If n is odd, then .

RATIONAL EXPONENTS & RADICALS

nn a a

11

nn a a

Properties of Radicals

Example: Use the properties to find the ff:

RATIONAL EXPONENTS & RADICALS

1) 24 6 24 6 4 6 6

12

2 22 6

2 22 6 2 6

12

Properties of Radicals

RATIONAL EXPONENTS & RADICALS

4

4

3242)

4

4

24

81 4

2

3

4 244

24

3 2

2

4 244 3 2

24 2

44 3

13

4 24

24

3 2

2

Simplifying of Radicals

A radical is simplified if the following hold:

RATIONAL EXPONENTS & RADICALS

1) There is no power in the radicand higher

than or equal to the index.

Examples: Simplify.

31) 48x 16 7 942) 16x y z33 16x

4 33 2 x 4 3x x4 2 342x yz y z

15

Simplifying of Radicals

A radical is simplified if the following hold:

RATIONAL EXPONENTS & RADICALS

2) The index and the exponents in the

radicand must have no common factor.

Examples: Simplify.

43) 4 2 6 464) r s t2

424 2 2

122 2

3 33 2 2 rs t s rt16

Simplifying of Radicals

A radical is simplified if the following hold:

RATIONAL EXPONENTS & RADICALS

3) There is no denominator in the radicand.

(There is no radical in the denominator.)

Examples: Simplify.

25)

3

x

y

6 3

7

86)

15

x y

z17

Simplifying of Radicals

RATIONAL EXPONENTS & RADICALS

Examples: Simplify.

25)

3

x

y 2

2 3 6

3 3 3

x y xy

y y y

2

6

3

xy

y

6

3

xy

y

18

Simplifying of Radicals

RATIONAL EXPONENTS & RADICALS

Examples: Simplify. 6 3

7

86)

15

x y

z

3 6 3

7

2

3 5

x y

z

3

3

2 2

3 5

x y y

z z

3

3

2 2 3 5

3 5 3 5

x y y z

z z z

3

3 2 2 2

2 30

3 5

x y yz

z z

3 3

4 4

2 30 230

15 15

x y yz x yyz

z z 19

Multiplying Radicals

Example: Find these products.

RATIONAL EXPONENTS & RADICALS

n n na b ab

3 24 41) 24 270x x

3 3 3 24 42 3 2 3 5x x

46 5x x42 3 5x x

4 4 54 2 3 5x

20

Multiplying Radicals

Examples: Find these products.

RATIONAL EXPONENTS & RADICALS

n n na b ab

32 2 2 2 2 23 32) 4 6 45x y x z x y z

6 4 3 3 3 6 4 33 34 6 45 2 3 5x y z x y z

2 36 5x yz y2 32 3 5x yz y

21

Multiplying Radicals

RATIONAL EXPONENTS & RADICALS

How do you multiply radicals with different

indices?

Examples: Find these products.

31) 2 211

322 23 2

6 62 2 1

63 22 2

6 5 62 32

Multiplying Radicals

RATIONAL EXPONENTS & RADICALS

2 432) x y z2 6 8 31 1

32 4 12 12 12x y z x y z

1126 8 3x y z

6 8 312 x y z

Dividing Radicals

Examples: Find these quotients.

, 0n

nn

a ab

bb

3 24

41)

3

x y

xy

3 2

41

3

x y

xy

2424

1

3 3

x yx y

RATIONAL EXPONENTS & RADICALS

Dividing Radicals

6 3

7

82)

15

x y

z

3 6 3

7

2

15

x y

z

RATIONAL EXPONENTS & RADICALS

3

3

2 2

15

x y y

z z

3

3

2 2 15

15 15

x y y z

z z z

3

3 2 2

2 30

15

x y yz

z z

3

3

2 30

15

x y yz

z z

3 3

4 4

2 30 230

15 15

x y yz x yyz

z z

Dividing Radicals

RATIONAL EXPONENTS & RADICALS

What if the indices are not equal?

Examples: Find these quotients.

3

21)

3

12

13

2

3

13 6

6

26

3

2

2 2

33

68

9

3 4 3 4

6 642 6

2 3 2 3

33 3

6 648

3

Dividing Radicals

RATIONAL EXPONENTS & RADICALS

34

32)

x

xy

34

13

x

xy

912

412

x

xy

9 9

1212 4 4 4

x x

x yxy

5

124

x

y

5 8 5 8

12 124 8 12

x y x y

y y y

5 812 x y

y

5 812

1212

x y

y

Rationalizing the Denominator

RATIONAL EXPONENTS & RADICALS

To rationalize the denominator means to get

rid of radicals in the denominator.

Multiply the numerator and denominator by

a rationalizing factor.

Rationalizing the Denominator

Example: Rationalize the denominator.

RATIONAL EXPONENTS & RADICALS

31)

2 3

3 2 3

2 3 2 3

2

2

3 2 3

2 3

6 3 3 6 3 3

4 3 1

6 3 3

Rationalizing the Denominator

RATIONAL EXPONENTS & RADICALS

22)

3 2 2 7

Rationalizing the Denominator

RATIONAL EXPONENTS & RADICALS

2 2

4 5 233)

x yz

x y z

Rationalizing the Denominator

RATIONAL EXPONENTS & RADICALS

4)a b

a b

a b a b

a b a b

2

2 2

a b

a b

2 2

2

a a b b

a b

2

a ab b

a b

Similar Radicals

Radicals are similar if they have the same

index and radicands when simplified.

RATIONAL EXPONENTS & RADICALS

Adding and Subtracting Radicals

We can only add (subtract) similar radicals.

To do that, add (subtract) their coefficients

and affix the common radical.

RATIONAL EXPONENTS & RADICALS

Adding and Subtracting Radicals

Examples: Find the following sums.

RATIONAL EXPONENTS & RADICALS

3 41) 48 12x x 2 3 2 43 4 3 2x x

24 3 2 3 (not similar)x x x

Adding and Subtracting Radicals

RATIONAL EXPONENTS & RADICALS

2632) 2 4x x 2 263 2 2x x

2

63 2 2x x

3 32 2x x 0

23 62 2x x

133 2 2x x

Adding and Subtracting Radicals

RATIONAL EXPONENTS & RADICALS

5 313) 8 2

2x x

x

3 5 31 22 2

2 2

xx x

x x

212 2 2 2

2x x x x x

x

321 1 4

2 2 2 2 22 2

xx x x x x x x

x x

Similar Radicals

Examples: Which of the following pairs of

radicals are similar?

RATIONAL EXPONENTS & RADICALS

3 41) 48 , 12x x

6 232) 2 , 4x x

513) , 8

2x

x

24 3 , 2 3x x x

3 32 , 2 x x

22, 2 2

2

xx x

x

SUMMARY

In this section, we learned

How to find the principal nth root of a

number

How to simplify radicals

RATIONAL EXPONENTS & RADICALS

SUMMARY

In this section, we learned

A radical is simplified if the following hold:

1. There is no power in the radicand higher than or equal to the index.

2. The index and the exponents in the radicand must have no common factor.

3. There is no denominator in the radicand.

RATIONAL EXPONENTS & RADICALS

SUMMARY

In this section, we learned

How to multiply and divide radicals

Radicals are similar if they have the same

index and radicand when simplified.

We can only add (subtract) similar radicals.

To do so, we add (subtract) their

coefficients and affix the common radical.

RATIONAL EXPONENTS & RADICALS

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