Name: Block:

Unit 4: Part 2

Complex Numbers

Day 1 Exponent Rules Day 2 Rational Exponents, Simplifying Square Roots,

The Imaginary Number, i

Day 3 Solving by Square Root Method

Day 4 Operations with Complex Numbers Day 5 Review Day 6 Quiz

Tentative Schedule of Upcoming Classes

Day 1 B Fri 11/6

Exponent Rules & Rational Exponents A Mon 11/9

Day 2 B Tues 11/10 Simplifying Square Roots

The Imaginary Number, i A Wed 11/11

Day 3 B Thurs 11/12

Solving by Square Root Method A Fri 11/13

Day 4 B Mon 11/16

Operations with Complex Numbers A Tues 11/17

Day 5 B Wed 11/18

Review* A Thurs 11/19

Day 6 B Fri 11/20

Quiz A Mon 11/23

*Skills Check #1 will be on Review Day!

Absent?

See Ms. Huelsman AS SOON AS POSSIBLE to get work and any help you need.

Notes are always posted online on the calendar. (If links are not cooperative, try changing to “list” mode)

You may also email Ms. Huelsman at Kelsey.huelsman@lcps.org with any questions!

____

Need Help?

Ms. Huelsman and Mu Alpha Theta are available to help Monday, Tuesday, Thursday, and Friday mornings in L506 starting at 8:10.

Ms. Huelsman is in L402 on Wednesday mornings.

Need to make up a test/quiz?

Math Make Up Room schedule is posted around the math hallway & in Ms. Huelsman’s classroom J

Overview of the rest of this unit… Where we’ve been –

Finding the vertex and graphing quadratics given the equation in standard, vertex, and intercept form.

Where we’re going – What does it mean to SOLVE a quadratic equation?

Finding the zeros without the aid of a calculator, given a quadratic. What we need to be able to do that –

An understanding of square roots and complex and imaginary numbers

Day 1: Exponent Rules

Today we will review simplifying expressions with integer exponents So we can extend this to fractional (rational) exponents

Does it really work? 2 • 4 =

2 • 4 = 21 • 22=

Practice: 51 • 52 =

x2 • x6 • x =

Does it really work? 26

=

(22)3 = (23)2 =

Practice: (25)3 =

(x2)4 =

Careful! ((-3)2)3 =

(-32)3 =

What’s different?

Does it really work? (2 x 3)2 = ( )2

Does this work on addition too? (2 + 3)2

=

COMMON MISTAKE: (x + 2)2

Use the property (2 x 3)2 =

Practice:

(4x)3 =

(9x4y)2 =

Careful! (-4z)2 =

–(4z)2 =

Does it really work? (Write the factors and cancel) 6

4

xx

=

Use the property…. 6

4

xx

=

14

8

bb

=

14

8

bb

=

Discussion: A preview… How does the quotient rule apply when the exponents are the same?

1 = 1616

= 2

2

44

= 1 = 4

4

xx

=

Write your own:

3

⎟⎟⎠

⎞⎜⎜⎝

⎛yx

27⎟⎠⎞⎜

⎝⎛−x

32

54

⎟⎟⎠

⎞⎜⎜⎝

⎛yx

52

⎟⎟⎠

⎞⎜⎜⎝

⎛ba

7x−=

0x6 =

52yx−

=

2x5 −

=

Watch this: 5

4

xx− There are two ways to simplify this problem:

a. Put everything where it belongs first (move neg. exponent): 9545

4

xxxxx ==−

OR

b. Use the Quotient of Powers rule: 954)5(45

4

xxxxx === +−−−

The first method usually creates the least amount of mistakes!!!! You try J

1. 2

6

xx−

2. 3

2

xx−

Things you need to MEMORIZE…

1. You must have the same base before using the rules! 2. 0x = 1 (anything to the zero power is 1)

3. nx− = nx1

means the reciprocal of nx .

Discuss: How would these exponent rules be different if the exponents were fractions?

Quick Questions: Which is correct?

Simplify A B 3 22 2• 54 52

2( 4)x+ ( 4)(x 4)x + + 2 24x + 016 1 0 2(3 )x −

29x−

2

19x

2

2

45xy

⎛ ⎞−⎜ ⎟⎝ ⎠

2

4

1625xy

− 2

4

1625xy

2

2

45xy

⎛ ⎞−⎜ ⎟⎝ ⎠

2

4

1625xy

− 2

4

1625xy

3

2

45xy

⎛ ⎞−⎜ ⎟⎝ ⎠

3

3

64125xy

− 3

3

64125xy

3

2

45xy

⎛ ⎞−⎜ ⎟⎝ ⎠

3

3

64125xy

− 3

3

64125xy

Day 2: Rational Exponents, Simplifying Square Roots & Imaginary Numbers

Today we will learn how to apply exponent rules to rational exponents and what those

mean; then we will learn to simplify square roots with both positive and negative radicands so that later we can solve quadratic functions!

First, Let’s review… Rules for Operations with Fractions Instructions Example

Multiplication

Multiply the numerators. Multiply the denominators. Simplify.

1 35 4× =

Division

Rewrite as a multiplication problem. Same (the first fraction is the same) Switch (the division sign à multiplication) Flip (the second fraction is flipped)

5 26 9÷ =

Addition Subtraction

Find a common denominator. Multiply the numerator and denominator by a number to create a common denominator. Add the numerators. Keep the same denominator.

3 27 3+ =

2 53 9− =

1) 1 29 9+ = 2) 1 2

9 9− = 3) 1 2

9 9• = 4) 4 1

9 3+ =

5) 4 19 3• = 6)

3 27 3÷ = 7) 3 2

7 3+ = 10) 3 2

7 3− =

11. Write a fraction addition problem in which the sum of 2 fractions equals 1. 12. Write a fraction multiplication problem in which the product of 2 fractions equals 1.

1. 3 12 25 5•

2.

12 23x

⎛ ⎞⎜ ⎟⎝ ⎠

3. 45

9

9−

4. (26 • 46)-1/6 = 5. 2/5

xx

=

Terminology & getting to know rational exponents

Exponential Notation

Radical Notation

     

     

     

    ( )2    or        

    ( )3    or      

am/n    

a-­‐m/n    

 

You  try:      Exponential Notation Radical Notation

   

 

     

     

   

 

     

 

Simplifying Radicals Using the Product Property (4.5):

a b a b• = • *a > 0, b > 0 What if the value is NOT a perfect square? This property helps us simplify radicals Perfect squares (integers): Perfect squares (exponents): Simplify these using the product rule. Assume all variables are positive.

3 24 200 381x

32 125 225x

  Addition and Subtraction: Must have _____________________________

3 4 3+ 7 3 12− 3 2 8− 12 7 3 4 5+ + Multiplication: Must have ___________________________________

2 8• = 7 6 2− • = 2 12 5 3• =

Product Property of Radicals  

Discuss: Which exponent property is being used when we multiply square roots? Division: Must have ___________________________________

520

182

2

169x

What do we do if the values cannot be divided??? Rationalize Denominators – multiply the denominator by a square root that will result in a perfect square under the radical (later we will extend this to other kinds of roots)

52

9 23

Discuss: Do the following “two’s” cancel?

2 62

Now let’s meet our new BFF…i…the “IMAGINARY NUMBER”

( )22

1

1 1

i

i

= −

= − = −  

i1 = ____ i2 = ____ i3 = ____ i4 = ____ i5 = ____ i6 = ____ i7 = ____ i8 = ____

Patterns of i: i9 = _____ i12 = _____ i21 = _____ i50 = _____ Using the patterns of i, how can we evaluate the expressions above quickly? Complex Number _________________ Pure Imaginary Number _________ where a is known as _____________ and b is known as _______________ Simplify:

81 8 24 ( )3− 2

81− 8− 24− 14−

   

Day 3: Solving Quadratics by Square Root Method

Today we will apply simplifying square roots with both positive and negative radicands So we can… Solve quadratic equations using the square root method.

Warm up: Review rational exponents and simplifying square roots

811/2 81−1/2

1811/2

181−1/2

−811/2 (−81)1/2

How are these last 2 different?

Applying simplifying square roots so we can SOLVE quadratics Using the “square root method”

1. x2 – 25 = 0

2. x2 + 25 = 0 What are the differences between #1 and #2?

3. 3x2 + 40 = -x2 + 56

−10 −8 −6 −4 −2 2 4 6 8 10

−10

−8

−6

−4

−2

2

4

6

8

10

x

y

−10 −8 −6 −4 −2 2 4 6 8 10

−10

−8

−6

−4

−2

2

4

6

8

10

x

y

−10 −8 −6 −4 −2 2 4 6 8 10

−10

−8

−6

−4

−2

2

4

6

8

10

x

y

4. 4(x – 1)2 = 8 What does this equation remind you of?

5. 4(x – 1)2 = - 8 How is this different from #4?

Discussion: What do you notice about ALL of these quadratic functions? What is a clue that we can use the square root method to solve a quadratic equation?

Write TWO quadratic equations that can be solved using the square root method. One should have two REAL solutions. The other should have two IMAGINARY (or complex) solutions.

−10 −8 −6 −4 −2 2 4 6 8 10

−10

−8

−6

−4

−2

2

4

6

8

10

x

y

−10 −8 −6 −4 −2 2 4 6 8 10

−10

−8

−6

−4

−2

2

4

6

8

10

x

y

Day 4: Complex Numbers – Now that we have them, what do we do with them?

Today we will define operations on complex numbers (and a few irrational numbers too!)

So we can…continue our investigation of quadratics and number systems. Recall….. Imaginary Unit _______ i1 = ____ i2 = ____ i3 = ____ i4 = ____ Complex Number _________________ Pure Imaginary Number _________ where a is known as _____________ and b is known as _______________ New Vocabulary….

Conjugates: (a b+ ) & (a b− ) Note: This is NOT multiplying by -1.

What happens to the radical? (a b+ )( a b− ) Complex Conjugates: (a + bi) & (a – bi) What happens to the imaginary number? (a + bi)(a – bi) Operations with Complex Numbers (Add, Subtract, Multiply, Divide)

**Remember that i is a NUMBER that behaves like a VARIABLE 1. (4 + 2i) + (-3 + 5i) 2. (-7 – 6i) + (9 + 8i)

3. (15 + 2i) – (18 + i)

4. (21 + 7i) – (3 – 5i)

5. 6i – (8 + 9i) + (3 – 4i) 6. (9 + 4i) (5 – 3i)

7. (-8 – 2i) (11 + 6i) 8. (4i) (13 – 7i)

9. 5i

10. 2 34ii+

11. 5 42 3

ii

++ 12.

2 63 4

ii

− +−

Discussion: How are operations with i similar/different from operations with square roots?

Using conjugates helps us out with radical problems, too!

1. 1255

2. 105

3. 5

2 3+ 4. 1

3 7−