Examples: Product Rule for Square Roots 6.2 – Simplified Form for Radicals.

40

Examples:

4 10

If and are real numbers, then a ba b a b Product Rule for Square Roots

2 10

7 75 7 25 3 7 5 3 35 3

6.2 – Simplified Form for Radicals

1716x xx1616 xx84

3 1716x 3 21528 xx 3 25 22 xx

104

3257

16

81

Examples:

2

5

4

9

45

49

aIf and are real numbers and 0, then

b

aa b b

b

Quotient Rule for Square Roots

2

25

9 5

7

3 5

7

16

81

2

25

45

49

6.2 – Simplified Form for Radicals

15

3

90

2

aIf and are real numbers and 0, then

b

aa b b

b

3 5

3

3 5

3

5

9 10

2

9 2 5

2

9 2 5

2

3 5

6.2 – Simplified Form for Radicals

Rationalizing the DenominatorRadical expressions, at times, are easier to work with if the denominator does not contain a radical. The process to clear the denominator of all radicals is referred to as rationalizing the denominator

5 3

5

x

5 31

5

x

5 53

5 5

x

5 15

25

x 5 15

5

x

15x

6.2 – Simplified Form for Radicals

5

3

7

20

5 3

3 3 5 3

3

7 20

20 20

7 20

20

7 4 5

20

2 35

20 35

10

7

20

7

4 5

7

2 5

7 5

2 5 5

35

2 25

35

2 5

35

10

6.2 – Simplified Form for RadicalsExamples:

2

45x

2

45x

2

9 5 x

2

3 5x

2 5

3 5 5

x

x x

10

3 5

x

x

10

15

x

x

6.2 – Simplified Form for RadicalsExamples:

Examples:

Theorem:6.2 – Simplified Form for Radicals

If “a” is a real number, then .

√ 40𝑥2

√ 4 ∙1 0 𝑥2

2|𝑥|√1 0

√𝑥2−16 𝑥+64

√ (𝑥− 8 )2

|𝑥− 8|

√18 𝑥3 − 9𝑥2

√9 𝑥2 (2 𝑥−1 )

3|𝑥|√2 𝑥−1

5 3x x

Review and Examples:

6 11 9 11

8x

15 11

12 7y y 5y

7 3 7 2 7

6.3 - Addition and Subtraction of Radical Expressions

27 75

Simplifying Radicals Prior to Adding or Subtracting

3 20 7 45

9 3 25 3

3 4 5 7 9 5

3 3 5 3 8 3

3 2 5 7 3 5

6 5 21 5 15 5

36 48 4 3 9 6 16 3 4 3 3

6 4 3 4 3 3 3 8 3

6.3 - Addition and Subtraction of Radical Expressions

Simplifying Radicals Prior to Adding or Subtracting

6 63 310 81 24p p

6 63 310 27 3 8 3p p

2 23 310 3 3 2 3p p

2 328 3p

2 23 330 3 2 3p p

6.3 - Addition and Subtraction of Radical Expressions

4 3 39 36x x x

Simplifying Radicals Prior to Adding or Subtracting

2 2 23 6x x x x x

23 6x x x x x

23 5x x x

6.3 - Addition and Subtraction of Radical Expressions

11x

Examples:

77

25

y

8

27

x

67

25

y y

3 7

5

y y

10x x 5x x

418x 49 2x 23 2x

8

9 3

x

4

3 3

x8

27

x

6.3 - Addition and Subtraction of Radical Expressions

3 88

Examples:

381

8

310

27

3

3

81

8

3 27 3

2

3 8 11 32 11

3 10

3

3

3

10

27

33 3

2

3 3 727m n 3 3 63 m n n 2 33mn n

6.3 - Addition and Subtraction of Radical Expressions

5 2

7 7

10 2x x

If and are real numbers, then a ba b a b

10

49 7

6 3 18 9 2 3 2

220x 24 5x 2 5x

6.4 –Multiplication and Division of Radical Expressions

Examples:

7 7 3 7 7 7 3 49 21

5 3 5x x

5 3x x

7 21

25 3 25x x 5 3 5x x

5 15x x

2 3 5 15x x x

𝑥−√3 𝑥+√5 𝑥−√15

6.4 –Multiplication and Division of Radical Expressions

Examples:

Review:

(x + 3)(x – 3) x2 – 3x + 3x – 9 x2 – 9

(√𝑥+3 ) (√𝑥− 3 )

√𝑥2−3 √𝑥+3√𝑥−9𝑥− 9

6.4 –Multiplication and Division of Radical Expressions

3 6 3 6

2

5 4x

9 6 3 6 3 36 3 36 33

5 4 5 4x x

225 4 5 4 5 16x x x

5 8 5 16x x

6.4 –Multiplication and Division of Radical Expressions

Examples:

2 5

2 1

22 5

2 11

1

2

4 2 5 2 5

4 2 2 1

2 6 2 5

2 1

7 6 2

1

7 6 2

If the denominator contains a radical and it is not a monomial term, then the use of a conjugate is required in order to rationalize the denominator.

conjugate

6.4 –Multiplication and Division of Radical Expressions

3

2 7

7

2 77

2

2

3

6 3 7

4 2 7 2 7 49

6 3 7

4 7

6 3 7

3

3 2 7

3

2 7 2 7

6.4 –Multiplication and Division of Radical Expressions

Example:

7

2 x

22

27 x

xx

2

7 2

4 2 2

x

x x x

7 2

4

x

x

6.4 –Multiplication and Division of Radical ExpressionsExample:

Radical Equations:

2 7x 6 1x x 9 2x

The Squaring Property of Equality:2 2, .If a b then a b

2 26, 6 .If x then x

2 25 2, 5 2 .If x y then x y

Examples:

6.5 – Equations Involving Radicals

Suggested Guidelines:

1) Isolate the radical to one side of the equation.

2) Square both sides of the equation.

3) Simplify both sides of the equation.

4) Solve for the variable.

5) Check all solutions in the original equation.

6.5 – Equations Involving Radicals

2 7x

2 2

2 7x

2 49x

51x

51 2 7

49 7

7 7

6.5 – Equations Involving Radicals

1

5x

6 1x x

2 2

6 1x x

6 1x x

5 1 0x

5 1x

1 16 1

5 5

6 11

5 5

6 5 1

5 5 5

1 1

5 5

6.5 – Equations Involving Radicals

9 2x

7x

2 2

7x

49x

49 9 2

7 9 2

16 2

no solution

6.5 – Equations Involving Radicals

52323 x

3323 x

333 332 x

2732 x

242 x

12x

5231223

523243

52273

523

55

6.5 – Equations Involving Radicals

3215 xx

115 xx

22115 xx

115 xxxx

1215 xxx

xx 224

01544 2 xx

01414 xx

01x

1x

014 x

22 224 xx

xxx 441616 2

042016 2 xx

4

1x

6.5 – Equations Involving Radicals

321115

32115

314

312

33

1x4

1x

324114

15

322114

5

323

41

323

21

32

6.5 – Equations Involving Radicals

1 5x x

1 5x x

2 2

1 5x x 21 10 25x x x

20 11 24x x

0 3 8x x 3 0 8 0x x

3 8x x

3 1 3 5

8 1 8 5

4 3 5 2 3 5

1 5

9 8 5

3 8 5 5 5

6.5 – Equations Involving Radicals

6.6 – Complex Numbers

1i

25 251 251 25i

Complex Number System:

This system of numbers consists of the set of real numbers and the set of imaginary numbers.

Imaginary Unit:

The imaginary unit is called i, where and .12 i

Square roots of a negative number can be written in terms of i.

i5

3 3i

32 32i 216i 24i

31

321

1i

72 72 ii 142i 14

The imaginary unit is called i, where and

.12 i

Operations with Imaginary Numbers

2

82

8i

2

24i

2

22i

125 ii 25 25i 5

327 327 i 81i i9

i2

6.6 – Complex Numbers

1iThe imaginary unit is called i, where and .12 i

Complex Numbers:

dicbia dibica idbca

idbca

Numbers that can written in the form a + bi, where a and b are real numbers.

3 + 5i 8 – 9i –13 + i

The Sum or Difference of Complex Numbers

dicbia cicbia dibica

6.6 – Complex Numbers

ii 3425 ii 3245 i9

342 i

ii 26 ii 26 i72

342 i i41

6.6 – Complex Numbers

ii 35 215i )( 115

ii 643

ii 262 2412 ii )1(412 i

2424318 iii )( 142118 i

Multiplying Complex Numbers

i124

i2122

6.6 – Complex Numbers

15

42118 i

ii 5656 225303036 iii )( 12536

Multiplying Complex Numbers

61

221 i 24221 iii

441 i i43

ii 2121

6.6 – Complex Numbers

)( 1441 i

2536

i3

ii

i

32

3ii

13i

Dividing Complex NumbersRationalizing the Denominator:

6.6 – Complex Numbers

ii

64

ii

ii

64

2

2

64iii

)( 1614

i

614

i

61

64

i

i3

32

61 i

i

i

32

3

i

i

i

i

32

32

32

3

2

2

9664

3296

iii

iii

94

3116

i

13

113 ii

13

11

13

3

Dividing Complex NumbersComplex Conjugates:The complex numbers (a + bi) and (a – bi) are complex conjugates of each other and,

(a + bi)(a – bi) = a2 + b2

6.6 – Complex Numbers

i

i

76

94

i

i

i

i

76

76

76

94

2

2

49424236

63542824

iii

iii

4936

638224

i

85

8239 ii

85

82

85

39

Dividing Complex NumbersComplex Conjugates:The complex numbers (a + bi) and (a – bi) are complex conjugates of each other and,

(a + bi)(a – bi) = a2 + b2

6.6 – Complex Numbers

i5

6i

i

i 5

5

5

6

225

30

i

i

25

30ii

5

6

Dividing Complex NumbersComplex Conjugates:The complex numbers (a + bi) and (a – bi) are complex conjugates of each other and,

(a + bi)(a – bi) = a2 + b2

6.6 – Complex Numbers

A Challenging Example

12 4 185 64x y z 10 2 4 15 35 32 2x x y z z 2 3 2 4 352 2x z x y z

2 𝑥2 𝑧3(2 𝑥2 𝑦4 𝑧3 )15

2 𝑥2 𝑧3215 𝑥

25 𝑦

45 𝑧

35

265 𝑥

1 25 𝑦

45 𝑧

185

6.3 - Addition and Subtraction of Radical Expressions