7.1, 7.2 & 7.3 Roots and Radicalsand

Rational Exponents• Square Roots, Cube Roots & Nth Roots

• Converting Roots/Radicals to Rational Exponents

• Properties of Exponents Apply to Rational Exponents Too!

•Simplifying Radical ExpressionsMultiplyingDividing

Try graphing : y = 3

x-1 And xy =

Square Roots & Cube Roots

A number b is a square root of a number a if b2 = a

25 = 5 since 52 = 25

Notice that 25 breaks down into 5 • 5So, 25 = 5 • 5

See a ‘group of 2’ -> bring it outside theradical (square root sign).

Example: 200 = 2 • 100 = 2 • 10 • 10 = 10 2

A number b is a cube root of a number a if b3 = a

8 = 2 since 23 = 8

Notice that 8 breaks down into 2 • 2 • 2 So, 8 = 2 • 2 • 2

See a ‘group of 3’ –> bring it outsidethe radical (the cube root sign)

Example: 200 = 2 • 100 = 2 • 10 • 10 = 2 • 5 • 2 • 5 • 2

= 2 • 2 • 2 • 5 • 5 = 2 25

3

3

3 3

3

3

3

3

Note: -25 is not a real number since nonumber multiplied by itself will be negative

Note: -8 IS a real number (-2) since-2 • -2 • -2 = -8

3

Nth Root ‘Sign’ Examples

16

-16

= 4 or -4

not a real number

-164

not a real number

Even radicals of negative numbersAre not real numbers.

-325

= -2 Odd radicals of negative numbersHave 1 negative root.

325

= 2 Odd radicals of positive numbersHave 1 positive root.

Even radicals of positive numbersHave 2 roots. The principal rootIs positive.

Exponent Rules( )x x

x x x

x

xx

m n mn

m n m n

m

nm n

x

xx

x x

mm

m m

0

1

1

1

/

(XY)m = xmym

XY

m

=Xm

Ym

Examples to Work through

3 34

4

3

8

12

81

27

yx

Product Rule and Quotient Rule Example

4/1

4/34/5

8

88

Some Rules for Simplifying Radical Expressions

nmn m

nn

nnn

aa

aa

abba

/

/1

Example Set 1

300

162

75

55

33

x

y

y

x

Example Set 2

4 4

3

512

54

16

x

Example Set 3

55

56

5

6

33

27

8

9

4

84

1255

r

t

r

t

tt

7.4 & 7.5: Operations on Radical Expressions

•Addition and Subtraction (Combining LIKE Terms)

•Multiplication and Division

• Rationalizing the Denominator

Radical Operations with Numbers

333 210545162

2423

Radical Operations with Variables

zzz

yxxy

xx

48312332

3

2

27

8

4 54 5

3

3

Multiplying Radicals (FOIL works with Radicals Too!)

)8)(9(

)32)(32(

xx

yxyx

Rationalizing the Denominator

• Remove all radicals from the denominator

3

2

1

y

xy

Rationalizing Continued…

• Multiply by the conjugate

23

3

23

1

7.6 Solving Radical Equations

25)63( 2 xX2 = 64

10003 x

1000)4( 3 x

#1

#2

#3

#4

Radical Equations Continued…

Example 1:

x + 26 – 11x = 4

26 – 11x = 4 - x

(26 – 11x)2 = (4 – x)2

26 – 11x = (4-x) (4-x)

26 - 11x = 16 –4x –4x +x2

26 –11x = 16 –8x + x2

-26 +11x -26 +11x0 = x2 + 3x -100 = (x - 2) (x + 5) x – 2 = 0 or x + 5 = 0 x = 2 x = -5

Example 2:

3x + 1 – x + 4 = 1

3x + 1 = x + 4 + 1

(3x + 1)2 = (x + 4 + 1)2

3x + 1 = (x + 4 + 1) (x + 4 + 1)

3x + 1 = x + 4 + x + 4 + x + 4 + 13x + 1 = x + 4 + 2x + 4 + 13x + 1 = x + 5 + 2x + 4 -x -5 -x -5 2x - 4 = 2x + 4 (2x - 4)2 = (2x + 4)2

4x2 –16x +16 = 4(x+4) 4x2 –20x = 0 4x(x –5) = 0, so…4x = 0 or x – 5 = 0 x = 0 or x = 5

4x+16

7.7 Complex Numbers

REAL NUMBERS Imaginary Numbers

IrrationalNumbers

, 8, -13

Rational Numbers(1/2 –7/11, 7/9, .33

Integers(-2, -1, 0, 1, 2, 3...)

Whole Numbers(0,1,2,3,4...)

Natural Numbers(1,2,3,4...)

Complex Numbers(a + bi)

Real Numbersa + bi with b = 0

Imaginary Numbersa + bi with b 0

i = -1 where

i2= -1

IrrationalNumbers

Rational Numbers

Integers

Whole Numbers

Natural Numbers

Simplifying Complex NumbersA complex number is simplified if it is in standard form:

a + bi

Addition & Subtraction)Ex1: (5 – 11i) + (7 + 4i) = 12 – 7i

Ex2: (-5 + 7i) – (-11 – 6i) = -5 + 7i +11 + 6i = 6 + 13i

Multiplication)Ex3: 4i(3 – 5i) = 12i –20i2 = 12i –20(-1) = 12i +20 = 20 + 12i

Ex4: (7 – 3i) (-2 – 5i) [Use FOIL] -14 –35i +6i +15i2

-14 –29i +15(-1) -14 –29i –15 -29 –29i

Complex ConjugatesThe complex conjugate of (a + bi) is (a – bi)The complex conjugate of (a – bi) is (a + bi)

(a + bi) (a – bi) = a2 + b2

Division7 + 4i2 – 5i

2 + 5i 14 + 35i + 8i + 20i2 14 + 43i +20(-1)2 + 5i 4 + 10i –10i – 25i2 4 –25(-1)

14 + 43i –20 -6 + 43i -6 434 + 25 29 29 29

= =

= + i=

Square Root of a Negative Number

25 4 = 100 = 10

-25 -4 = (-1)(25) (-1)(4)

= (i2)(25) (i2)(4) = i 25 i 4 = (5i) (2i) = 10i2 = 10(-1) = -10

Optional Step

Practice – Square Root of Negatives

i 1

12

16

4

Practice – Simplify Imaginary Numbers

i2 =

i3 =

i4 =

i5 =

i6 =

-1

-i

1

i

-1

i0 = 1i1 = i

Another way to calculate in

Divide n by 4. If the remainder is rthen in = ir

Example:i11 = __________

11/4 = 2 remainder 3

So, i11 = i3 = -i

Practice – Simplify More Imaginary Numbers

203

100

26

15

i

i

i

i

Practice – Addition/Subtraction

)7()93(

)7()93(

ii

ii 10 +8i

-4 +10i

Practice – Complex Conjugates

• Find complex conjugate.

i

i

43

25

3i =>

-4i =>

Practice Division w/Complex Conjugates

i

i

4

47

4__2i

=

Things to Know for Test

1. Square Root, Cube Root, Nth Root - Simplify

2. Rational Exponents – Convert back and forth to/from radical form

3. Add, Subtract, Multiply & Divide radicals & rational exponents

4. Rationalize denominator

5. Solve radical equations

6. Imaginary Numbers – Add, subtract, multiply, divide

7. Imaginary Numbers – find the value of in