Chapter 7Chapter 7Radicals, Radicals,

Radical Functions, Radical Functions, and Rational and Rational ExponentsExponents

7.1 Radical Expressions and 7.1 Radical Expressions and FunctionsFunctions

Square RootSquare Root If a >= 0, If a >= 0,

then b, where b >= 0, such that bthen b, where b >= 0, such that b22 = a, = a, is the principal square root of ais the principal square root of a

√ √ a = ba = b E.g.,E.g.,

√√25 = 525 = 5 √√100 = 10100 = 10

4 2 2 4 2 2 22 4 4

---- = ----, because --- = ----- ---- = ----, because --- = ----- 49 7 7 49 49 7 7 49

9 + 16 = 25 = 59 + 16 = 25 = 5

9 + 16 = 3 + 4 = 79 + 16 = 3 + 4 = 7

Negative Square RootNegative Square Root

25 = 5 ---- principal square root25 = 5 ---- principal square root - 25 = -5 ---- negative square root- 25 = -5 ---- negative square root Given: aGiven: a

What is the square root of a?What is the square root of a? Given: 25Given: 25

What is the square root of 25?What is the square root of 25? sqrt = 5, sqrt = -5, because 5sqrt = 5, sqrt = -5, because 522 = 25, (-5) = 25, (-5)22

= 25= 25 Note. in - a = b, a must be >= 0Note. in - a = b, a must be >= 0

Your TurnYour Turn

Find the value of x:Find the value of x:

1.1. x = 121x = 121 x = 11, -11x = 11, -11

2.2. x = 3 + 13x = 3 + 13 x = 4, -4x = 4, -4

3.3. x = 36/81x = 36/81 x = 6/9 = 2/3, -2/3x = 6/9 = 2/3, -2/3

Square Root FunctionSquare Root Function

f(x) = xf(x) = x

xx y = xy = x (x, y)(x, y)

00 00 (0,0)(0,0)

11 11 (1,1)(1,1)

44 22 (4,2)(4,2)

99 33 (9,3)(9,3)

1616 44 (16,4)(16,4)

1818 4.244.24 (18,4.2)(18,4.2)x

y

Excel Chart

Evaluating Evaluating a Square Root Functiona Square Root Function

Given: f(x) = 12x – 20Given: f(x) = 12x – 20 Find: f(3)Find: f(3) Solution:Solution:

f(3) = 12(3) – 20f(3) = 12(3) – 20 = 36 – 20 = 36 – 20 = 16 = 16 = 4 = 4

Domain of a Square Root Domain of a Square Root FunctionFunction

Given: f(x) = 3x + 12Given: f(x) = 3x + 12 Find the Domain of f(x):Find the Domain of f(x): Solution:Solution:

Radicand must be non-zero.Radicand must be non-zero.

3x + 12 ≥ 03x + 12 ≥ 03x ≥ -123x ≥ -12x ≥ -4x ≥ -4

[-4, ∞)[-4, ∞)

Radicand

ApplicationApplication By 2005, an “hour-long” show on prime time By 2005, an “hour-long” show on prime time

TV was 45.4 min on the average, and the TV was 45.4 min on the average, and the rest was commercials, plugs, etc. But this rest was commercials, plugs, etc. But this amount of “clutter“ was leveling off in recent amount of “clutter“ was leveling off in recent years. The amount of non-program years. The amount of non-program “clutter”, in minutes, was given by:“clutter”, in minutes, was given by:

M(x) = 0.7 x + 12.5M(x) = 0.7 x + 12.5

where x is the number of years after 1996.where x is the number of years after 1996. What was the number of minutes of “clutter” What was the number of minutes of “clutter”

in an hour program in 2002? in an hour program in 2002?

SolutionSolution

Solution:Solution: M(x) = 0.7 x + 12.5M(x) = 0.7 x + 12.5 x = 2002 – 1996 = 6x = 2002 – 1996 = 6

M(6) = 0.7 6 + 12.5M(6) = 0.7 6 + 12.5 ~ 0.7(2.45) + 12.5 ~ 0.7(2.45) + 12.5 ~ 14.2 (min) ~ 14.2 (min)

In 2009?In 2009? x = 2009 x = 2009 –– 1996 = 13 1996 = 13 M(13) = 0.7 13 + 12.5M(13) = 0.7 13 + 12.5

~ 15 (min) ~ 15 (min)

Cube Root and Cube Root Cube Root and Cube Root FunctionFunction

a = b,a = b, means bmeans b33 = a = a

8 = 2, 8 = 2, because 2because 233 = 8 = 8

-64 = -4-64 = -4 Because (-4)Because (-4)33 = -64 = -64

3

3

3

Cube Root FunctionCube Root Function f(x) = xf(x) = x3

xx y = xy = x (x, y)(x, y)-27-27 -3-3 (-27,-(-27,-

3)3)

-8-8 -2-2 (-8,-2)(-8,-2)

-1-1 -1-1 (-1,-1)(-1,-1)

00 00 (0,0)(0,0)

11 11 (1,1)(1,1)

88 22 (8,2)(8,2)

2727 33 (27,3)(27,3)

3030 3.13.1 (30,3.1(30,3.1))

3

Simplifying Radical Simplifying Radical ExpressionsExpressions

-64x-64x33 = (-4x) = (-4x)33 = -4x = -4x 81 = (3)81 = (3)44 = 3 = 3 -81 = x has no solution in R,-81 = x has no solution in R,

since there is no x such that xsince there is no x such that x44 = -81 = -81

In generalIn general -a -a hashas an an nthnth root when n is odd root when n is odd -a -a has no has no nthnth root when n is even root when n is even

3

3 4

4

4

n n

Your TurnYour Turn

Simplify the following:Simplify the following:

1.1. 17 + 1917 + 19

2.2. 4 · 25 4 · 25

3.3. (-2)(2)(-2)(2)

4.4. 33 125 125

5.5. 33 -27 -27

1. ±62. ±103. No solution4. 55. -3

7.2 Rational Exponents7.2 Rational Exponents

What is the meaning of What is the meaning of 771/31/3?? x = 7x = 71/31/3 meansmeans

xx33 = (7 = (71/31/3))33 = 7 = 7 Generally, aGenerally, a1/n1/n is number such that is number such that

(a(a1/n1/n))nn = a = a

Your TurnYour Turn SimplifySimplify

1.1. 64641/21/2

2.2. (-125)(-125)1/31/3

3.3. (6x2y)(6x2y)1/31/3

4.4. (-8)(-8)1/31/3

SolutionsSolutions1.1. 882.2. -5-53.3. 6x2y6x2y4.4. -2-2

3

SolveSolve

100010002/32/3

= (1000= (10001/31/3))22 = 10 = 1022 = 100 = 100 16163/23/2

(16(161/21/2))33 = 4 = 433 = 64 = 64 -32-323/53/5

-(32-(321/51/5))33 = -(2) = -(2)33 = -8 = -8

Negative ExponentNegative Exponent

What is the meaning of the What is the meaning of the following? following? 55-2-2

We want 5We want 5-2-2 · 5 · 522 = 5 = 51 1 = 5= 5Thus, 5Thus, 5-2-2 = 1/5 = 1/522

ExamplesExamplesn 33-3-3 = 1/3 = 1/333 = 1/27 = 1/27n 55-3-3 = 1/5 = 1/53 3 = 1/125= 1/125n 99-1/2-1/2 = 1/9 = 1/91/21/2 = 1/3 = 1/3

Order of PrecedenceOrder of Precedence

What is the differenceWhat is the difference between between -32-323/53/5 and and (-32) (-32) 3/5 3/5

Note: 2Note: 255 = 32 = 32

between between -16-163/43/4 and and (-16)(-16) 3/4 3/4

Note: 2Note: 244 = 16 = 16

SimplifySimplify

661/71/7 ·· 6 64/7 4/7

= 6= 6(1/4 + 4/7) (1/4 + 4/7) = 6= 65/75/7

32x32x1/21/2

16x16x3/43/4

= 2x= 2x(1/2 (1/2 –– 3/4) 3/4) = 2x = 2x-1/4-1/4

(8.3(8.33/43/4))2/32/3

= 8.3= 8.3(3/4 ∙(3/4 ∙ 2/3)2/3)

= 8.3= 8.31/21/2

SimplifySimplify 4949-1/2-1/2

= (7= (722))-1/2 -1/2 == 77-1-1 = 1/7 = 1/7 (8/27)(8/27)-1/3-1/3

= 1/(8/27)= 1/(8/27)1/31/3 = (27/8) = (27/8)1/31/3 = 27 = 271/31/3/8/81/31/3 = 3/2 = 3/2 (-64)(-64)-2/3-2/3

= 1/(-64)= 1/(-64)2/32/3 = 1/((-64) = 1/((-64)1/31/3))22 = 1/(-4) = 1/(-4)22 = 1/16 = 1/16 (5(52/32/3))33

= 5= 52/32/3 ∙ 3 ∙ 3 = 5 = 522 = 25 = 25 (2x(2x1/21/2))55

2255xx1/2 1/2 ·· 5 5 = 32x = 32x5/25/2

7.3 Multiplying & 7.3 Multiplying & SimplifyingSimplifying

Radical ExpressionsRadical Expressions Product RuleProduct Rule

a a ·· b = ab or b = ab or aa1/n1/n ·· b b1/n 1/n = (ab) = (ab)1/n1/n

Note: Factors have same order of root.Note: Factors have same order of root. E.g,E.g,

50 2 = 50 50 2 = 50 ·· 2 = 100 = 10 2 = 100 = 10 2000 = 400 2000 = 400 ·· 5 = 400 5 = 400 ·· 5 = 20 5 = 20

5 5

nnn

Simplify Radicals by Simplify Radicals by FactoringFactoring

√ √(80)(80) = √(8 · 2 · 5) = √(2= √(8 · 2 · 5) = √(233 · 2 · 5) · 2 · 5)

= √(2= √(244 · 5) = 4√(5) · 5) = 4√(5) √√(40)(40)

= √(8 · 5) = √(2= √(8 · 5) = √(233 · 5) · 5)= 2√(5)= 2√(5)

√ √(200x(200x44yy22)) = √(5 · 40x= √(5 · 40x44yy22) = √(5 · 5 · 8x) = √(5 · 5 · 8x44yy22))

= √(5= √(522 · 2 · 222 · 2x · 2x44yy22) ) = 5 · = 5 · 22xx22y√(2) = 10xy√(2) = 10x22y√(2) y√(2)

3

3 3

3

√ √(80)(80) = √(8 · 2 · 5) = √(2= √(8 · 2 · 5) = √(233 · 2 · 5) · 2 · 5)

= √(2= √(244 · 5) = 4√(5) · 5) = 4√(5) √√(40)(40)

= √(8 · 5) = √(2= √(8 · 5) = √(233 · 5) · 5)= 2√(5)= 2√(5)

√ √(200x(200x44yy22)) = √(5 · 40x= √(5 · 40x44yy22) = √(5 · 5 · 8x) = √(5 · 5 · 8x44yy22))

= √(5= √(522 · 2 · 222 · 2x · 2x44yy22) ) = 5 · = 5 · 22xx22y√(2) = 10xy√(2) = 10x22y√(2) y√(2)

Simplify Radicals by Simplify Radicals by FactoringFactoring

√√(64x(64x33yy77zz2929)) = √(32 = √(32 ·· 2x 2x33yy55yy22zz2525zz44))= √(2= √(255yy55zz2525 ·· 2x 2x33yy22zz44))= 2yz= 2yz55√(2x√(2x33yy22zz44))

5

5

5

5

Multiplying & Multiplying & SimplifyingSimplifying

√√(15)(15)··√(3)√(3) = √(45) = √(9= √(45) = √(9··5) = 3√(5)5) = 3√(5)

√√(8x(8x33yy22))··√(8x√(8x55yy33)) = √(64x= √(64x88yy55) = √(16) = √(16··4x4x88yy44y)y)= 2x= 2x22y√(4y)y√(4y)

4 4

4 4

4

ApplicationApplication

Paleontologists use the functionPaleontologists use the function W(x) = 4 W(x) = 4√(2x)√(2x)to estimate the walking speed of a to estimate the walking speed of a dinosaur, W(x), in feet per second, dinosaur, W(x), in feet per second, where x is the length, in feet, of the where x is the length, in feet, of the dinosaurdinosaur’’s leg. s leg. What is the walking speed of a What is the walking speed of a dinosaur whose leg length is 6 feet?dinosaur whose leg length is 6 feet?

W(x) = 4W(x) = 4√(2x)√(2x) W(6) = 4√(2W(6) = 4√(2··6)6) = 4√(12) = 4√(12) = 4√(4 = 4√(4··3)3) = 8√(3) = 8√(3) ~~ 8·(1.7) 8·(1.7) ~ ~ 14 (ft/sec)14 (ft/sec)

(humans: 4.4 ft/sec walking(humans: 4.4 ft/sec walking 22 ft/sec running) 22 ft/sec running)

Your TurnYour Turn

Simplify the radicalsSimplify the radicals √√(2x/3)(2x/3)··√√(3/2)(3/2) = √((2x/3)(3/2)) = √x= √((2x/3)(3/2)) = √x √√(x/3)(x/3)··√√(7/y)(7/y) = √((x/3)(7/y)) = √(7x/3y)= √((x/3)(7/y)) = √(7x/3y) √√(81x(81x88yy66)) = √(27= √(27··3x3x66xx22yy66)= 3x)= 3x22yy22√(3x√(3x22)) √√((x+y)((x+y)44)) =√((x+y)=√((x+y)33(x+y))= (x+y)√(x+y)(x+y))= (x+y)√(x+y)

4 4

3

3

3 3

3 3

4 4

7.4 Adding, Subtracting, & 7.4 Adding, Subtracting, & DividingDividing

Adding ( Adding (radicals with same indices & radicals with same indices & radicandsradicands)) 8√(13) + 2√(13)8√(13) + 2√(13)

= √(13) = √(13) ·· (8 + 2) = 10√(13) (8 + 2) = 10√(13) 7√(7) – 6x√(7) + 12√(7)7√(7) – 6x√(7) + 12√(7)

= √(7) = √(7) ··(7 – 6x + 12) = (19 – 6x)√(7)(7 – 6x + 12) = (19 – 6x)√(7) 7√(3x) - 2√(3x) + 2x7√(3x) - 2√(3x) + 2x22√(3x)√(3x)

= √(3x)= √(3x)··(7 – 2 + 2x(7 – 2 + 2x22)) = (5 += (5 + 2x2x22) √(3x) ) √(3x)

3 3 3

3 3

4 4 4

4 4

Adding Adding 7√(18) + 5√(8)7√(18) + 5√(8)

= 7√(9= 7√(9··2) + 5√(42) + 5√(4··2) = 72) = 7··3 √(2) + 53 √(2) + 5··2√(2)2√(2)= 21√(2) + 10√(2) = 31√(2)= 21√(2) + 10√(2) = 31√(2)

√√(27x) - 8√(12x)(27x) - 8√(12x) = √(9= √(9··3x) - 8√(43x) - 8√(4··3x) = 3√(3x) – 83x) = 3√(3x) – 8··2√(3x)2√(3x)

= √(3x)= √(3x)··(3 – 16) = -13√(3x)(3 – 16) = -13√(3x) √√(xy(xy22) + √(8x) + √(8x44yy55))

= √(xy= √(xy22) + √(8x) + √(8x33yy33xyxy22) = √(xy) = √(xy22) + 2xy √(xy) + 2xy √(xy22) ) = √(xy= √(xy22) (1 + 2xy)) (1 + 2xy)= (1 + 2xy) √(xy= (1 + 2xy) √(xy22) )

3 3

3 3 3 3

3

3

Dividing Radical Dividing Radical ExpressionsExpressions

Recall: (a/b)Recall: (a/b)1/n 1/n = (a)= (a)1/n1/n/(b)/(b)1/n1/n

(x(x22/25y/25y66))1/21/2

=(x=(x22))1/2 1/2 // (25y (25y66))1/21/2

=x/5y=x/5y33

((45xy45xy))1/21/2/(/(2·52·51/21/2)) = (1/2) = (1/2) ··(45xy/5)(45xy/5)1/21/2 = (1/2) ·(9·5xy/5) = (1/2) ·(9·5xy/5)1/21/2

= (1/2) ·3(xy)= (1/2) ·3(xy)1/21/2 = (3/2) = (3/2) ··(xy)(xy)1/21/2

(48x(48x77y)y)1/31/3/(6xy/(6xy-2-2))1/31/3

= ((48x= ((48x77y)/6xyy)/6xy-2-2))))1/31/3

= (8x= (8x66yy33))1/31/3

= 2x= 2x22yy

7.5 Rationalizing 7.5 Rationalizing DenominatorsDenominators

Given: 1Given: 1 √(3) √(3)Rationalize the denominator—get rid Rationalize the denominator—get rid of the radical in the denominator.of the radical in the denominator.

1 √(3) √(3) 1 √(3) √(3) = = √(3) √(3) 3 √(3) √(3) 3

Denominator Containing Denominator Containing 2 Terms2 Terms

Given: 8Given: 8 3√(2) + 4 3√(2) + 4

Rationalize denominatorRationalize denominator Recall: (A + B)(A – B) = ARecall: (A + B)(A – B) = A22 – B – B22

8 3√(2) – 4 8(3√(2) – 4) 8 3√(2) – 4 8(3√(2) – 4) = =3√(2) + 4 3√(2) – 4 (3√(2) )3√(2) + 4 3√(2) – 4 (3√(2) )22 – (4) – (4)22

24 √(2) - 32 8(3 √(2) – 4) 12 √(2) - 24 √(2) - 32 8(3 √(2) – 4) 12 √(2) - 1616 = = = = 18 – 16 2 18 – 16 2

Your TurnYour Turn

Rationalize the denominatorRationalize the denominator 2 + √(5) 2 + √(5)

√(6) - √(3) √(6) - √(3) 2+√(5) √(6)+√(3) 2+√(5) √(6)+√(3)

2√(6)+2√(3)+√(5)√(6)+√(5)√(3) 2√(6)+2√(3)+√(5)√(6)+√(5)√(3) = = √(6) - √(3) √(6)+√(3) 6 – 3 √(6) - √(3) √(6)+√(3) 6 – 3

2√(6) + 2√(3) + √(30) +√(15) 2√(6) + 2√(3) + √(30) +√(15) = = 3 3

7.6 Radical Equations7.6 Radical Equations

ApplicationApplication A basketball player’s hang time is A basketball player’s hang time is

the time in the air while shooting a the time in the air while shooting a basket. It is related to the vertical basket. It is related to the vertical height of the jump by the following height of the jump by the following formula:formula: t = √(d) / 2 t = √(d) / 2

A Harlem Globetrotter slam-dunked A Harlem Globetrotter slam-dunked while he was in the air for 1.16 while he was in the air for 1.16 seconds. How high did he jump?seconds. How high did he jump?

Solving Radical Solving Radical EquationsEquations

√√(x) = 10(x) = 10 (√(x))(√(x))22 = 10 = 1022

x = 100x = 100 √√(2x + 3) = 5(2x + 3) = 5

(√(2x + 3) )(√(2x + 3) )22 = = 5522

(2x + 3) = 25(2x + 3) = 252x = 222x = 22x = 11x = 11

Check

√√(2x + 3) = 5(2x + 3) = 5 √(2(11) + 3) = 5 ? √(2(11) + 3) = 5 ? √(22 + 3) √(22 + 3) = 5 ? √(25) = 5 ?√(25) = 5 ? 5 = 5 5 = 5 yesyes

SolveSolve

√√(x - 3) + 6 = (x - 3) + 6 = 55 √√(x - 3) = -1(x - 3) = -1

(√(x - 3))2 = (-(√(x - 3))2 = (-1)21)2(x – 3) = 1(x – 3) = 1x = 4x = 4

Check:√√(x - 3) + 6 = 5(x - 3) + 6 = 5 √(4 - 3) + 6 = 5 ? √(4 - 3) + 6 = 5 ? √(1) + 6 = 5 ? √(1) + 6 = 5 ? 1 + 6 = 5 ? 1 + 6 = 5 ? FalseFalseThus, there is no Thus, there is no solution to this solution to this equation.equation.

Your TurnYour Turn

Solve: √(x – 1) + Solve: √(x – 1) + 7 = 27 = 2 √√(x – 1) = -5(x – 1) = -5

(√(x – 1))(√(x – 1))22 = (-5) = (-5)22

x – 1 = 25x – 1 = 25x = 26x = 26

Check:

√√(x – 1) + 7 = 2(x – 1) + 7 = 2√(26 – 1) + 7 = 2 ?√(26 – 1) + 7 = 2 ?√(25) + 7 = 2 ?√(25) + 7 = 2 ?5 + 7 = 2 ? 5 + 7 = 2 ? False False

Thus, there is no Thus, there is no solution to this solution to this equation.equation.

Your TurnYour Turn Solve: x + √(26 – 11x) Solve: x + √(26 – 11x)

= 4= 4 √√(26 – 11x) = 4 – x(26 – 11x) = 4 – x

(√(26 – 11x))(√(26 – 11x))22 = (4 – x) = (4 – x)22

26 – 11x = 16 – 8x + x26 – 11x = 16 – 8x + x22

0 = x0 = x22 + 3x – 10 + 3x – 10xx22 + 3x – 10 = 0 + 3x – 10 = 0(x – 2)(x + 5) = 0(x – 2)(x + 5) = 0x – 2 = 0x – 2 = 0x = 2x = 2x + 5 = 0x + 5 = 0x = -5x = -5

Check -5:

√√(26 – 11x) = 4 – x(26 – 11x) = 4 – x√(26 – 11(-5)) = 4 – (-√(26 – 11(-5)) = 4 – (-5) ?5) ?√(26 + 55) = 4 + 5 √(26 + 55) = 4 + 5 ? ?√(81) = 9 √(81) = 9 ? ?9 = 9 9 = 9 TrueTrue

Check 2:Check 2:

√√(26 – 11x) = 4 – x(26 – 11x) = 4 – x√(26 – 11(2)) = 4 – 2 √(26 – 11(2)) = 4 – 2 ??√(4) = 2 √(4) = 2 ? ?2 = 2 2 = 2 TrueTrue

Solution: {-5, 2}Solution: {-5, 2}

Hang Time in BasketballHang Time in Basketball

A basketball player’s hang time is the A basketball player’s hang time is the time spent in the air when shooting a time spent in the air when shooting a basket. It is a function of vertical height basket. It is a function of vertical height of jump.of jump.

√(d) √(d)t = ----- where t is hang time in sec and t = ----- where t is hang time in sec and 2 d is vertical distance in feet. 2 d is vertical distance in feet.

If Michael Wilson of Harlem If Michael Wilson of Harlem Globetrotters had a hang time of 1.16 Globetrotters had a hang time of 1.16 sec, what was his vertical jump?sec, what was his vertical jump?

Hang TimeHang Time

√ √(d)(d)t = ----- t = ----- 2 2

2t = √(d)2t = √(d)2(1.16) = √(d)2(1.16) = √(d)2.32 = √(d)2.32 = √(d)(2.32)(2.32)22 = (√(d)) = (√(d))22

5.38 = d5.38 = d

7.7 Complex Numbers7.7 Complex Numbers

What kind of number is x = √(-25)?What kind of number is x = √(-25)? xx22 = -25? = -25?

Imaginary Unit Imaginary Unit ii i = i = √(-1), √(-1), i i 22 = -1 = -1

ExampleExample √√(-25) = √((25)(-1)) = √(25)√(-1) = 5(-25) = √((25)(-1)) = √(25)√(-1) = 5ii √√(-80) = √((80)(-1)) = √((16 · 5)(-1)) (-80) = √((80)(-1)) = √((16 · 5)(-1))

= 4√(5) = 4√(5)ii = 4 = 4i i √(5)√(5)

Your TurnYour Turn

Express the following with Express the following with i.i.1.1. √√(-49)(-49)

2.2. √√(-21)(-21)

3.3. √√(-125)(-125)

4.4. -√(-300)-√(-300)

Complex NumbersComplex Numbers

Comlex number has a Real part and Comlex number has a Real part and an Imaginary part of the form: a + an Imaginary part of the form: a + bbii

ExampleExample1.1. 2 + 32 + 3ii

2.2. -4 + 5-4 + 5ii

3.3. 5 – 25 – 2ii

Adding and Subtracting Adding and Subtracting Complex NumbersComplex Numbers

(5 – 11i) + (7 + 4i)(5 – 11i) + (7 + 4i)= 5 – 11i + 7 + 4i= 5 – 11i + 7 + 4i= 12 – 7i= 12 – 7i

(2 + 6i) – (12 – 4i)(2 + 6i) – (12 – 4i)= 2 + 6i – 12 + 4i= 2 + 6i – 12 + 4i= -10 + 10i= -10 + 10i

Multiplying Complex Multiplying Complex NumbersNumbers

4i(3 – 5i)4i(3 – 5i)= 12i – 20i= 12i – 20i22

= 12i – 20(-1)= 12i – 20(-1)= 12 + 12i= 12 + 12i

(5 + 4i)(6 – 7i)(5 + 4i)(6 – 7i)= 5·6 – 5 ·7i + 4i· 6 – 4 ·7i= 5·6 – 5 ·7i + 4i· 6 – 4 ·7i22

= 30 – 35i + 24i – 28(-1)= 30 – 35i + 24i – 28(-1)= 30 – 11i + 28= 30 – 11i + 28= 58 – 11i= 58 – 11i

MultiplyingMultiplying

1.1. √√(-3) √(-5)(-3) √(-5)= i√(3) · i√(5)= i√(3) · i√(5)= i= i22 √(15) √(15)= -√(15)= -√(15)

2.2. √√(-5) √(-10)(-5) √(-10)= i√(5) · i√(10)= i√(5) · i√(10)= i= i22 √(50) √(50)= -√(50)= -√(50)= -√(25 · 2)= -√(25 · 2)= -5√(2)= -5√(2)

Conjugates and DivisionConjugates and Division Given: a + bGiven: a + bii

Conjugate of a + bi: Conjugate of a + bi: a – ba – biiConjugate of a – bi: Conjugate of a – bi: a + ba + bii

Why conjugates?Why conjugates?(a + bi)(a – bi) (a + bi)(a – bi) = (a)= (a)22 – (bi) – (bi)22

= a= a22 – b – b22ii22

= a= a22 + b + b22 (3 + 2i)(3 – 2i) = 9 – (2i)(3 + 2i)(3 – 2i) = 9 – (2i)22= 9 – 4(-1) = 13= 9 – 4(-1) = 13 Multiplying a complex number by its Multiplying a complex number by its

conjugate results in a real number.conjugate results in a real number.

Dividing Complex Dividing Complex NumbersNumbers

Express 7 + 4iExpress 7 + 4i -------- as a + bi -------- as a + bi 2 – 5i 2 – 5i

7 + 4i (7 + 4i) (2 + 5i) 14 + 35i + 7 + 4i (7 + 4i) (2 + 5i) 14 + 35i + 8i + 208i + 20-------- = ---------- · ----------- = -------------------------------- = ---------- · ----------- = ------------------------2 – 5 i (2 – 5i) (2 + 5i) 4 + 252 – 5 i (2 – 5i) (2 + 5i) 4 + 25

34 – 43i 34 – 43i= -------------= ------------- 29 29

Your TurnYour Turn

6 + 2i6 + 2i----------------4 – 3i4 – 3i

6 + 2i (4 + 3i) 24 + 18i + 8i 6 + 2i (4 + 3i) 24 + 18i + 8i + 6i+ 6i22

= ---------- · ---------- = -------------------------= ---------- · ---------- = ------------------------- (4 – 3i) (4 + 3i) 16 + 9 (4 – 3i) (4 + 3i) 16 + 9 (18 + 26i) (18 + 26i)= -------------= ------------- 25 25

Your TurnYour Turn

5i – 45i – 4-------------- 3i 3i

(5i – 4) -3i -15i(5i – 4) -3i -15i22 + 12i + 12i= --------- · ----- = -------------------= --------- · ----- = ------------------- 3i -3i -9i 3i -3i -9i22

15 + 12i 3(5 + 4i) 5 + 4i 15 + 12i 3(5 + 4i) 5 + 4i= ------------ = ----------- = ---------= ------------ = ----------- = --------- 9 9 3 9 9 3

Powers of Powers of ii

ii22 = -1 = -1ii33 = (-1) = (-1)i = -ii = -iii44 = (-1) = (-1)22 = 1 = 1ii55 = ( = (ii44)i)i = = iiii66 = (-1) = (-1)33 = -1 = -1ii77 = ( = (ii66))ii = = -i-iii88 = (-1) = (-1)44 = 1 = 1ii99 = ( = (ii88))ii = = iiii1010 = (-1) = (-1)55 = -1 = -1

Your TurnYour Turn

SimplifySimplify ii1717

ii1717 = i = i1616i = (ii = (i22))88i = ii = i ii5050

ii5050 = (i = (i22))2525 = (-1) = (-1)2525 = -1 = -1 ii3535

ii3535 = (i = (i3434)i = (i)i = (i22))1717i = (-1)i = (-1)1717i = -ii = -i

ApplicationApplication

Electrical engineers use the Ohm’s law to Electrical engineers use the Ohm’s law to relate the current (I, in amperes), voltage relate the current (I, in amperes), voltage (E, in volts), and resistence (R, in ohms) (E, in volts), and resistence (R, in ohms) in a circuit:in a circuit:

E = IRE = IR Given: I = (4 – 5i) and R = (3 + 7i), what Given: I = (4 – 5i) and R = (3 + 7i), what

is E?is E? E = (4 – 5i)(3 + 7i) = 12 + 28i - 15i - 35iE = (4 – 5i)(3 + 7i) = 12 + 28i - 15i - 35i22

= 47 + 13i (volts) = 47 + 13i (volts)