1.5.a radicals

RATIONAL EXPONENTS & RADICALS

2

OBJECTIVES

Upon completion, you should be able to

Simplify radicals; and

Perform addition, subtraction, multiplication and division of radicals.


RADICALS

A radical (or irrational expression) is an

algebraic expression involving rational

exponents. It is of the form:


n

mn m aa

3

Examples

Write as a radical:


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:


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:


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


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 .


nn a a

11

nn a a


Example: Use the properties to find the ff:


1) 24 6 24 6 4 6 6

12

2 22 6

2 22 6 2 6

12



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:


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




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




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



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



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.


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


Examples: Find these products.


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



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



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


Dividing Radicals

6 3

7

82)

15

x y

z

3 6 3

7

2

15

x y

z


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


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


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


To rationalize the denominator means to get

rid of radicals in the denominator.

Multiply the numerator and denominator by

a rationalizing factor.


Example: Rationalize the denominator.


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



22)

3 2 2 7



2 2

4 5 233)

x yz

x y z



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.


Adding and Subtracting Radicals

We can only add (subtract) similar radicals.

To do that, add (subtract) their coefficients

and affix the common radical.



Examples: Find the following sums.


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

24 3 2 3 (not similar)x x x



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



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?


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


SUMMARY



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.


SUMMARY


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.


Engineering

1.5.a radicals