5 3x x

Review and Examples:

6 11 9 11

8x

15 11

12 7y y 5y

7 3 7 2 7

7.4 – Adding, Subtracting, Multiplying 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

7.4 – Adding, Subtracting, Multiplying Radical Expressions

4 3 39 36x x x

Simplifying Radicals Prior to Adding or Subtracting

6 63 310 81 24p p

2 2 23 6x x x x x

23 6x x x x x 23 5x x x

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

7.4 – Adding, Subtracting, Multiplying 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

7.4 – Adding, Subtracting, Multiplying Radical Expressions

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

7.4 – Adding, Subtracting, Multiplying Radical Expressions

𝑥−√3 𝑥+√5 𝑥−√15

3 6 3 6

25 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

7.4 – Adding, Subtracting, Multiplying Radical Expressions

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 radical is referred to as rationalizing the denominator

5 35

x

5 3 15

x

5 535 5

x

5 1525

x 5 15

5x

15x

7.5 – Rationalizing the Denominator of Radicals Expressions

53

720

5 33 3

5 33

7 2020 20

7 20

20

7 4 5

20

2 35

20 35

10

720

7

4 5

7

2 5

7 52 5 5

35

2 25

352 5

3510

7.5 – Rationalizing the Denominator of Radicals Expressions

245x

2

45x

29 5 x

23 5x

2 53 5 5

xx x

10

3 5xx

1015

xx

7.5 – Rationalizing the Denominator of Radicals Expressions

7.5 – Rationalizing the Denominator of Radicals Expressions

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

Review:

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

(x + 7)(x – 7) x2 – 7x + 7x – 49 x2 – 49

2 52 1

22 52 11

12

4 2 5 2 54 2 2 1

2 6 2 52 1

7 6 2

1

7 6 2

7.5 – Rationalizing the Denominator of Radicals Expressions

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

conjugate

32 7

72 77

22

3

6 3 7

4 2 7 2 7 49

6 3 74 7

6 3 73

3 2 7

3

2 7 2 7

7.5 – Rationalizing the Denominator of Radicals Expressions

conjugate

72 x

22

27 xxx

2

7 2

4 2 2

x

x x x

7 2

4

x

x

conjugate

7.5 – Rationalizing the Denominator of Radicals Expressions

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:

7.6 – Radical Equations and Problem Solving

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.

7.6 – Radical Equations and Problem Solving

2 7x

2 22 7x

2 49x

51x

51 2 7

49 7

7 7

7.6 – Radical Equations and Problem Solving

15

x

6 1x x

2 26 1x x

6 1x x

5 1 0x

5 1x

1 16 15 5

6 115 5

6 5 15 5 5

1 15 5

7.6 – Radical Equations and Problem Solving

9 2x

7x

2 27x

49x

49 9 2

7 9 2

16 2

no solution

7.6 – Radical Equations and Problem Solving

7.6 – Radical Equations and Problem Solving

52323 x

3323 x

333 332 x

2732 x

242 x

12x

5231223

523243

52273

523

55

7.6 – Radical Equations and Problem Solving

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

41

x

7.6 – Radical Equations and Problem Solving

321115

32115

314

312

33

1x41

x

324114

15

324114

5

347

41

347

21

349

1 5x x

1 5x x

2 21 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

7.6 – Radical Equations and Problem Solving

7.7 – 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

7.7 – Complex Numbers1i

72 72 ii 142i 14

The imaginary unit is called i, where and

.12 i

Operations with Imaginary Numbers

28

28i

224i

222i

125 ii 25 25i 5

327 327 i 81i i9

i2

7.7 – Complex Numbers1iThe 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

7.7 – Complex Numbers

ii 3425 ii 3245 i9

342 i

ii 26 ii 26 i72

342 i i41

7.7 – Complex Numbers

ii 35 215i 15

ii 643

ii 262 2412 ii 412 i

2424318 iii 42118 i

Multiplying Complex Numbers

i124

i2122

7.7 – Complex Numbers

ii 5656 225303036 iii 2536

Multiplying Complex Numbers

61

221 i 24221 iii

441 i i43

ii 2121

7.7 – Complex Numbers

ii32

3

ii

ii

3232

323

2

2

96643296iiiiii

94

3116

i

13113 i

i1311

133

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

7.7 – Complex Numbers

ii

7694

ii

ii

7676

7694

2

2

4942423663542824

iiiiii

4936638224

i

858239 i

i8582

8539

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

7.7 – Complex Numbers

i56

ii

i 55

56

22530

ii

2530i

i56

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