martin-gay, developmental mathematics 1 warm-up #5

Martin-Gay, Developmental Mathematics 1

Warm-Up #5

1. Find the product of ab. .

2. Simplify

3. Estimate: What is


Homework

Advanced: Simplifying Radical Worksheet

Page 1. #1-6

. #1-6

Regular: Simplifying Radical Worksheet

Page 1. #1-4

Page 2. #1-4

Introduction to Radicals


The principal (positive) square root is noted as

a

The negative square root is noted as

a

Principal Square Roots


Perfect Squares

1

4

916

253649

64

81

100121

144169196

225

256

324

400

625

289


16

25

100

144

= 4 or -4

= 5 or -5

= 10 or -10

= 12 or -12


20

32

75

40

= =

=

=

5*4

2*16

3*25

10*4

=

=

=

=

52

24

35

102

Perfect Square Factor * Other Factor

LE

AV

E I

N R

AD

ICA

L F

OR

M


The cube root of a real number a

abba 33 ifonly

Example:

Cube Roots

8)2)(2)(2(2 because 28 33


Cube Roots

3 27

A cube root of any positive number is positive.

Examples:

3 5

43

125

64

3 8 2

A cube root of any negative number is negative.

3 a

15.1 – Introduction to Radicals

3 27 3 3 8 2


3 27 3

3 68x 22x

Cube Roots

Example

Simplifying Radicals


baab

0b if b

a

b

a

a bIf and are real numbers,

Product Rule for Radicals


Simplify the following radical expressions.

40 104 102

16

5 16

5

4

5

15 No perfect square factor, so the radical is already simplified.


Example



7x xx6 xx6 xx3

16

20

x

16

20

x

8

54

x 8

52

x


Example


nnn baab

0 if n

n

n

n bb

a

b

a

n a n bIf and are real numbers,

Quotient Rule for Radicals



3 16 3 28 33 28 3 2 2

3

64

3 3

3

64

3

4

33


Example

Adding and Subtracting Radicals


Sums and Differences

Rules in the previous section allowed us to split radicals that had a radicand which was a product or a quotient.

We can NOT split sums or differences.

baba

baba


What is combining “like terms”?

Similarly, we can work with the concept of “like” radicals to combine radicals with the same radicand.

Like Radicals


373 38

24210 26

3 2 42 Can not simplify

35 Can not simplify

Adding and Subtracting Radical Expressions

Example


Simplify the following radical expression. 331275

3334325

3334325

333235

3325 36

Example



Simplify the following radical expression.

91464 33

9144 3 3 145

Example



Simplify the following radical expression. Assume that variables represent positive real numbers.

xxx 5453 3 xxxx 5593 2

xxxx 5593 2

xxxx 5533

xxxx 559

xxx 59 xx 510

Example


Multiplying and Dividing Radicals


nnn abba

0 if b b

a

b

an

n

n

n a n bIf and are real numbers,

Multiplying and Dividing Radical Expressions



xy 53 xy15

23

67

ba

ba

23

67

ba

ba44ba 22ba

Multiplying and Dividing Radical Expressions

Example


If we rewrite the expression so that there is no radical in the denominator, it is called rationalizing the denominator.

Rationalizing the denominator is the process of eliminating the radical in the denominator.

Rationalizing the Denominator


Rationalize the denominator.

2

3

2

2

3 9

6

3

3

3

3

22

23

2

6

33

3

39

3 6

3

3

27

3 6

3

3 6 33 3 2


Example


Many rational quotients have a sum or difference of terms in a denominator, rather than a single radical.

•need to multiply by the conjugate of the denominator

•The conjugate uses the same terms, but the opposite operation (+ or ).

Conjugates

32

23

15

23


Rationalize the denominator.

32

23

332322

3222323

32

32

32

322236

1

322236

322236


Example

martin-gay, developmental mathematics 1 warm-up #5

Documents

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