Rationalizing the Denominator Radical expressions, at times, are easier to work with if the denominator does not contain a radical. The process to clear.

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 3

5

x

5 31

5

x

5 53

5 5

x

5 15

25

x 5 15

5

x

15x

7.5 – Rationalizing the Denominator of Radicals Expressions

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

7.5 – Rationalizing the Denominator of Radicals Expressions

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

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 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

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

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

7.5 – Rationalizing the Denominator of Radicals Expressions

conjugate

7

2 x

22

27 x

xx

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 2

2 7x

2 49x

51x

51 2 7

49 7

7 7

7.6 – Radical Equations and Problem Solving

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

7.6 – Radical Equations and Problem Solving

9 2x

7x

2 2

7x

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

4

1x

7.6 – Radical Equations and Problem Solving

321115

32115

314

312

33

1x4

1x

324114

15

324114

5

347

41

347

21

34

9

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

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 Numbers

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

7.7 – 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

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

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 Numbers

Complex 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

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 Numbers

Complex 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

i5

6i

i

i 5

5

5

6

225

30

i

i

25

30ii

5

6

Dividing Complex Numbers

Complex Conjugates:

The complex numbers (a + bi) and (a – bi) are complex conjugates of each other and,

(a + bi)(a – bi) = a2 + b2