Martin-Gay, Developmental Mathematics 1

Warm-Up #6 (Thursday, 9/17)

1. Simplify

2. Simplify

3. Simplify

Martin-Gay, Developmental Mathematics 2

Warm-Up #7 (Friday, 9/18)

1. Simplify

2. Simplify

3. Simplify

Martin-Gay, Developmental Mathematics 3

Homework (Thursday, 9/17)

Lesson 1.03_pg 14

Advanced: #1, 2, 9, 10

Regular: #1, #2

Martin-Gay, Developmental Mathematics 4

Homework (Friday, 9/18)

Advanced: make sure to have warm-up #4-7

Regular: make sure to have warm-up #4-7

Lesson 1.03_pg 14_#1 and #2

Martin-Gay, Developmental Mathematics 5

The cube root of a real number a

abba 33 ifonly

Example:

Cube Roots

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

Simplifying Radicals

Martin-Gay, Developmental Mathematics 7

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

Martin-Gay, Developmental Mathematics 8

3 27 3

3 68x 22x

Cube Roots

Example

Martin-Gay, Developmental Mathematics 9

baab

0b if b

a

b

a

a bIf and are real numbers,

Product Rule for Radicals

Martin-Gay, Developmental Mathematics 10

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.

Simplifying Radicals

Example

Martin-Gay, Developmental Mathematics 11

Simplify the following radical expressions.

7x xx6 xx6 xx3

16

20

x

16

20

x

8

54

x 8

52

x

Simplifying Radicals

Example

Martin-Gay, Developmental Mathematics 12

nnn baab

0 if n

n

n

n bb

a

b

a

n a n bIf and are real numbers,

Quotient Rule for Radicals

Martin-Gay, Developmental Mathematics 13

Simplify the following radical expressions.

3 16 3 28 33 28 3 2 2

3

64

3 3

3

64

3

4

33

Simplifying Radicals

Example

Adding and Subtracting Radicals

Martin-Gay, Developmental Mathematics 15

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

Martin-Gay, Developmental Mathematics 16

What is combining “like terms”?

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

Like Radicals

Martin-Gay, Developmental Mathematics 17

373 38

24210 26

3 2 42 Can not simplify

35 Can not simplify

Adding and Subtracting Radical Expressions

Example

Martin-Gay, Developmental Mathematics 18

Simplify the following radical expression. 331275

3334325

3334325

333235

3325 36

Example

Adding and Subtracting Radical Expressions

Martin-Gay, Developmental Mathematics 19

Simplify the following radical expression.

91464 33

9144 3 3 145

Example

Adding and Subtracting Radical Expressions

Martin-Gay, Developmental Mathematics 20

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

Adding and Subtracting Radical Expressions

Multiplying and Dividing Radicals

Martin-Gay, Developmental Mathematics 22

nnn abba

0 if b b

a

b

an

n

n

n a n bIf and are real numbers,

Multiplying and Dividing Radical Expressions

Martin-Gay, Developmental Mathematics 23

Simplify the following radical expressions.

xy 53 xy15

23

67

ba

ba

23

67

ba

ba44ba 22ba

Multiplying and Dividing Radical Expressions

Example

Martin-Gay, Developmental Mathematics 24

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

Martin-Gay, Developmental Mathematics 25

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

Rationalizing the Denominator

Example

Martin-Gay, Developmental Mathematics 26

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

Martin-Gay, Developmental Mathematics 27

Martin-Gay, Developmental Mathematics 28

Rationalize the denominator.

32

23

332322

3222323

32

32

32

322236

1

322236

322236

Rationalizing the Denominator

Example