Algebra 1 Unit 6 Notes Name:____________________________________

NOTE: You should be prepared for daily quizzes.

Every student is expected to do every assignment for the entire unit, or else Homework

Club will be assigned!

Student who complete 100% of the assignments for the semester will receive a 2% bonus!

HW reminders:

If you cannot solve a problem, get help before the assignment is due.

Need help? Try www.khanacademy.org or www.classzone.com

Day Date Assignment (Due the next class meeting) Monday

Tuesday

01/13/14 (A)

01/14/14 (B)

6.1 Multiplying Exponents

Wednesday

Thursday

01/15/14 (A)

01/16/14 (B)

6.2 Dividing Exponents

Friday

Tuesday

01/17/14 (A)

01/21/14 (B))

6.3 Negative Exponents

Wednesday

Thursday

01/22/14 (A)

01/23/14 (B)

6.4 Simplifying Radicals

Friday

Monday

01/24/14 (A)

01/27/14 (B)

6.5 Product Property of Radicals

Tuesday

Wednesday

01/28/14 (A)

01/29/14 (B)

6.6 Simplifying Radicals with Multiple Terms

Thursday

Friday

01/30/14 (A)

01/31/14 (B)

6.7 Cube Roots

Monday

Tuesday

02/03/14 (A)

02/04/14 (B)

Practice Test

Wednesday

Thursday

02/05/14 (A)

02/06/14 (B)

Unit 6 Test

6.1 Multiplying Exponents

Essential Question: How do we simplify expressions by applying exponent

properties involving products?

Warm-up:

1. Graph 𝐲 = 𝟑 2. Graph 𝐲 = −𝟏 3. Graph 𝐲 = −𝟒

With your teacher, graph each line over the specified domain:

1. If x < 2 2. If x ≥ 0 3. If -1 ≤ x < 7

Vocabulary:

What does 𝑥4 mean? What does 𝑥3 mean?

Simplify the following expression, 𝑥4 ∙ 𝑥3.

Coefficient: 4𝑥3 𝑥

Base: 4𝑥3 𝑥

Exponent: 4𝑥3 𝑥

Product-of-Powers Property: For all nonzero numbers x and all integers m and n,

𝑥𝑚 ∙ 𝑥𝑛 = 𝑥(______+______)

***Note: This only works when the bases are the same.

When multiplying two terms with the same _____________, we keep the base and __________

the exponents.

Examples: Simplify each of the following expressions. Leave in exponential form.

1. 25 ∙ 28 2. 𝑥3 ∙ 𝑥6 3. 𝑦4 ∙ 𝑦

4. 𝑧7 ∙ 3𝑧9 5. (𝑥2𝑦3)(𝑥3𝑦6) 6. 2𝑦2𝑧4 ∙ 3𝑦9𝑧6

Objective #1: Can you find the product of terms with the same base?

a. 𝑥3 ∙ 𝑥8 ∙ 𝑥 b. 4𝑎5 ∙ 6𝑎7 c. 𝑦3𝑧2 ∙ 𝑥𝑦5𝑧2

Reflect: What are 2 different ways to write out the expression 𝑥3𝑦𝑥5𝑦7in order to simplify it?

What does (𝒙𝟒)𝟑 mean? Simplify (𝒓𝟓)𝟐.

Power-of-a-Power Property: For all nonzero number x and all integers m and n,

(𝑥𝑚)𝑛 = 𝑥(____∙____)

Examples: Simplify each of the following expressions. Leave in exponential form.

1. (𝑥2)3 2. (𝑦5)4 3. (23)2

Objective #2: Can you simplify an expression to more than one exponent?

a. (34)2 b. (𝑥5)2 c. – (𝑥)3

What does (𝑟𝑠)3 mean?

Power-of-a-Product Property: For all nonzero numbers x and y, and any integer n,

(𝑥𝑦)𝑛 = 𝑥___𝑦___

Examples: Simplify each of the following expressions. Leave in exponential form.

1. (𝑥𝑦)3 2. (3𝑏)5 3. (−4𝑥)2

Objective #3: Can you simplify a product to an exponent?

a. (3𝑥)4 b. (𝑥2𝑧3)5 c. (−5𝑎4)2

Example: Now try using all three properties in the same problem.

1. (3𝑥𝑦2)3 ∙ 𝑥2 2. (−5𝑣𝑤4)2 ∙ 2𝑣𝑤3

Objective #4: Can you use all three properties in the same problem?

a. (4𝑎3𝑏2)2(3𝑎) b. (−4𝑟5𝑠)2 ∙ 2𝑟𝑠3

Objective #5: Can you explain and fix the mistakes made in the following

problems?

a. (𝑦3)8 = 𝑦11

b. 2𝑤5 ∙ 3𝑤2 ∙ (−3𝑤2)4 = 6𝑤7 ∙ −9𝑤8 = −54𝑤56

6.2 Dividing Exponents

Essential Question: How do we simplify expressions by applying exponent

properties involving quotients?

Warm Up:

1. Graph 𝒚 = 𝒙 + 𝟏 2. Graph 𝐲 = −𝟏

𝟐𝒙 − 𝟐 3. Graph 𝐲 = 𝟐𝐱 − 𝟑

With your teacher, graph each line over the specified domain:

1. If x < 2 2. If x ≥ 0 3. If -1 ≤ x < 5

What does 24

22 mean?

Quotient-of-Powers: For all nonzero numbers a and any positive integers m and n, m > n,

𝑎𝑚

𝑎𝑛 = 𝑎(_____−_____)

***Note: This only works when the bases are the same.

When dividing two terms with the same __________, we keep the base and __________ the

exponents.

Examples: Simplify each of the following expressions.

1. 612

65= 6______−______ = 6_____ 2.

42∙48

44=

4____

44= 4_____−_____ = 4____

3. (−2)7

(−2)4 4.

1

𝑦9∙ 𝑦12 5.

𝑥6

𝑥3∙𝑥2

Objective #6: Can you find the quotient of two terms with the same base?

a. ℎ5

ℎ2 b.

(−5)23

(−5)17 c.

𝑚3𝑚5

𝑚7

Reflect: Can you explain the difference between simplifying a product with the same base

and simplifying a quotient with the same base?

What does (3

4)

2 mean?

Power-of-a-Quotient Property: For all real numbers a and b, b≠0, and a positive integer m,

(𝑎

𝑏)

𝑚=

𝑎_____

𝑏_____ , 𝑏 ≠ 0

Examples: Simplify each expression.

1. (4

7)

3=

4___

7___ 2. (𝑟

𝑠)

5=

3. (−4

𝑤)

3 4. (

35∙34

33 )2

5. (5

𝑡)

4 6. (

2𝑦7

𝑥𝑦5)3

7. (3𝑥2

𝑤)

2

∙2

3𝑤

Objective #7: Can you simplify a quotient to an exponent?

a. (5

9)

2 b. (

𝑥

𝑦)

5 c. (

3𝑟3

𝑟2 )4

d. (𝑥2𝑦3

𝑦2 )3

e. 1

4𝑥3∙ (

2𝑥5

𝑦)

3

6.3 Zero and Negative Exponents

Essential Question: How do we simplify expressions by defining and using zero

and negative exponents?

Warm Up:

1. Graph 𝒚 = −𝟑𝒙 𝒊𝒇 𝒙 > −𝟐 2. Graph 𝐲 =𝟏

𝟑𝒙 − 𝟑 𝒊𝒇 − 𝟔 ≤ 𝒙 ≤ 𝟑

Definition of Zero and Negative Exponents

a to the ______ power 𝑎0 = ___ , 𝑎 ≠ 0 50 = __

𝑎−𝑛 is the reciprocal of 𝑎𝑛

𝑎−𝑛 =1

𝑎𝑛 , 𝑎 ≠ 0 2−1 = __

𝑎𝑛 is the reciprocal of 𝑎−𝑛

1

𝑎−𝑛 = 𝑎𝑛 , 𝑎 ≠ 0 1

2−1 = ___

***Remember, when dividing two terms with the same base, we subtract the exponents.

𝑥3 1 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥

𝑥2

𝑥1 = 𝑥

𝑥0

𝑥−1

𝑥−2

Examples: 𝑎6

𝑎4 =𝑎∙𝑎∙𝑎∙𝑎∙𝑎∙𝑎

𝑎∙𝑎∙𝑎∙𝑎=

𝑎∙𝑎

1= ____ or

𝑎6

𝑎4 = 𝑎6−4 = ____

𝑎6

𝑎6 =

= = ____ or 𝑎6

𝑎6 = 𝑎_____ = 𝑎____ = ___

𝑎4

𝑎6 =

=

= or 𝑎4

𝑎6 = 𝑎______ = 𝑎____ =

Examples: Simplify each quotient. Be sure all exponents are positive.

1. 𝑥3

𝑥7 2. 𝑥3

𝑥3 3. 𝑥−6

4. (−2)−2 5. 1

𝑥−9 6. 𝑥5 ∙ 𝑥−5

7. 𝑥−2

𝑥−4 8. 𝑦−2

𝑥5

Special Cases:

9. (1

4)

−3 10. 0−7

Example: Simplify the following expressions. Write your answer using only positive

exponents.

1. 2𝑤−3𝑥

(2𝑤𝑥)2 2. 6𝑓𝑔−4

2𝑓2𝑔 3. (3𝑦𝑧2)−2

Reflect: Summarize the order in which you should simplify when there are exponents in a

problem?

Objective #8: Can you simplify expressions involving zero and negative

exponents?

a. −30 b. 𝑥6 ∙ 4𝑥−6 c. (5𝑛6

𝑛2𝑛4)3

d. 4−3 e. 𝑥

𝑥−4 f. (

2

5)

−3

6.4: Simplifying Radical Expressions

Essential Question: Can you simplify expressions that involve radicals?

Warm Up:

1. Simplify (𝟐𝒙𝟐𝒚𝟓)𝟐 2. Graph 𝐲 = −𝟐

𝟑𝒙 𝒊𝒇 − 𝟔 ≤ 𝒙 < 𝟗

Squaring a number means multiplying a number by ______________.

The numbers 1, 4, 9, 16 are examples of ____________________ _________________,

because 1 = 12,

4 = 22, 9 = 32, 16 = 42, etc.

Make a list of perfect squares below (up to 225):

The inverse of squaring is finding a ______________________ ___________________.

Example 1: Simplify each expression.

a) √49 b) √64

c) A square television set has an area of 144 square inches. Find the length of one side.

Estimating Radical Expressions:

What does the expression √36 mean? _____________ We are trying to find a number that we

can __________________ ____ ____________ to equal 36.

Sometimes the numbers inside a radical, √ , will not be perfect squares. When this happens we

need to __________________ the value of the expression.

Examples: Estimate the value of each radical expression.

1. √42 2. √102

3. √75 4. √155

5. A square piece of paper has an area of 80 square inches. Which of the following is the best

estimate for the length of one side of the paper?

A) 9.1 inches C) 40 inches

B) 8.9 inches D) 39 inches

Objective #9: Can you estimate the value of a radical expression?

a. √60 b. √8 c. √201

**d. Which integer is closest to the value of √53?

A. 3

B. 8

C. 11

D. 15

**e. The value of 3√180 is between what two integers?

A. 13 and 14

B. 39 and 42

C. 60 and 61

D. 23 and 24

Simplest Form of a Radical Expression: A radical expression is in simplest form if:

a) no __________________ _____________ are factors of the inside.

b) no _________________ are in the _____________________ of a fraction.

Product Property of Radicals: The square root of a product equals the _____________ of the

____________ __________ of the factors.

√ab = √a ∙ √b Example: √20 = √4 ∙ √5 = 2√5

***Note that there are no factors of 5 that are perfect squares.

Examples: Simplify each of the following radical expressions.

1. √12 2. √50 3. √200

4. √180 5. √88 6. √20 ∙ 75

Objective #10: Can you simplify radical expressions?

a. √24 b. √90 c. √72

**d. √23 ∙ 23 **e. √50 ∙ 25

f. Describe the error in the problem below. Then work the problem correctly.

√45 = √3 ∙ 15 = √3 ∙ 3 ∙ 5 = 5√3

Radical Expressions involving Variables:

What does √x2 mean? What does √x3 mean?

Example: Simplify √12𝑥2

Examples: Simplify each of the radical expressions.

1. √48𝑥3 2. √12𝑥2𝑦5 3. 2√8𝑧4

Objective #11: Can you simplify radical expressions involving

variables?

a. √𝑥8 b. √h3 c. √x ∙ x2

**d. √x3 ∙ x3 **e. √50𝑥3𝑦5

Reflect: Find √𝑥37 …without writing out 37 x’s.

6.5: Product Property of Radicals

Essential Question: Can you simplify the product and quotients of radical

expressions?

Warm up:

1) Simplify: 𝒙𝟑 ∙ 𝒙𝟓 2) Simplify:

𝒄𝟔

𝒄𝟐

3) Given the system of equations {𝒙 + 𝟑𝒚 = 𝟗

−𝒚 = −𝟐𝒙 + 𝟒 What is the solution for y in the

system?

Product Property of Radicals: The product of two radicals equals the _____________ of the

____________.

√𝑎 ∙ √𝑏 = √ab Example: √2 ∙ √8 = √16 = 4

Examples: Find the product and simplify each radical expression.

1. √3 ∙ √3 2. √10 ∙ √2 3. √12 ∙ √3

4. √7(3√21) 5. √𝑥5 ∙ √𝑥 6. 3√6 ∙ 4√2

Objective #12: Can you simplify products of radical expressions?

a. √6 ∙ √8 b. √5(6√7) c. 2√x3 ∙ 3√x5

d. x√2 ∙ √8 **e. (√25𝑥)(√4𝑥3)

f. Describe the error in the problem below. Then work the problem correctly.

√15 ∙ √5 = √20 = √4 ∙ 5 = √2 ∙ 2 ∙ 5 = 2√5

Quotient Property of Radicals: The square root of a quotient equals the ____________ of the

___________ __________ of the numerator and denominator.

√a

b=

√a

√b where a ≥ 0 and b > 0.

1. Reduce the fraction

2. Square root the top and bottom

Examples: Simplify each of the radical expressions.

1. √25

49 2. √

3

12 3. √

14𝑥3

18𝑥2

Objective #13: Can you simplify quotients of radical expressions?

a. √49

16 b. √

8

9 c. √

20

5

d. √50

40 e. √

20

18 **f. √

12𝑥3

4𝑥2

****Note: All of the examples above simplified to a whole number on the denominator.

Sometimes that will not be the case. When the denominator is not a whole number, we have to

“rationalize the denominator.”

Examples: Simplify each radical expression and be sure to rationalize the denominator.

1. √5

7 2. √

9

18

3. √16𝑎3

3𝑎4 4. √15𝑥3

12 5. √

5

60

Rationalizing the Denominator: The process of ___________________ a ____________ from

an expression’s denominator.

√4

11 =

2

√11 We cannot leave the radical in the _________________, so we must

rationalize!

1. Reduce the fraction

2. Simplify any radicals

3. Rationalize the

Denominator

Objective #14: Can you simplify quotients of radical expressions and

rationalize the denominator?

a. √2

3 b. √

12

5 c. √

16

8

d. √3

24 e. √

20𝑏3

14𝑏 **f. √

12

11𝑏

Reflect: How do you know when to rationalize the denominator of a radical expression?

6.6 Simplifying Radical Expressions with more than one term

Essential Question: Can you simplify radical expressions using the distributive

property and by combining like terms?

Warm up:

Simplify

1) 3𝑥2 + 5𝑥 − 𝑥 2) 4𝑦 − 2𝑥 + 6𝑦 − 3

3) √252 4) √3

24

Combining Like Terms:

If two or more expressions have the same radicand, the stuff inside the square root, then we can

combine like terms just like variables.

Example: 5√3 + 12√3

We can also simplify radicals to possibly create like terms.

Example: 7√2 - 4√12 + 6√8

Simplify each of the following radical expressions.

1. 12√7 - 10√2 - 4√7 2. √9 + √20 − √45

3. √200 − 3√32 + 6 4. 2√6 + 3√600

1. Simplify each term

2. Combine like terms

Objective #15: Can you combine like terms when radical expressions

are involved?

a. 2√3 + 4√5 + 6√3 b. √23 + 5√11 + 4√23 − 7√11

c. √75 + √27 d. √7 + 7√28 − 2√63

Reflect: Define “like terms” for radical expressions.

Distributing Radical Terms: We can use the ___________________ property with radicals!

Example: √2(3 − 4√5)

Simplify each of the following expressions.

1. √3(4√2 − √3) 2. √7(6√2 + 2√14)

3. √2x(√3 + √3x) 4. √5𝑥(√x-√5)

Objective #16: Can you distribute with radical expressions?

a. 5(√3 + √12) b. √5(2√2 − √5) + 12

6.7 Cube roots and simplifying radicals

Essential Question: Can you estimate cube roots and simplify problems involving

radicals?

Warm up:

Estimate the value of the following radicals:

1) √30 2) √70

2) Simplify 42 − 20 + [(5 − 2)2 + 3]

Cubing a number means multiplying it by itself ___________ times.

The inverse of cubing a number is finding a _____________ ______________.

Cubes Equal Factors Cube Root

23 = 8 2 ∙ 2 ∙ 2 √83

= √2 ∙ 2 ∙ 23

= 2

33 =

43 =

53 =

63 =

Example: Estimate which two whole numbers a cube root is between.

1) √203

2) √803

3) √1003

Example: Order the expressions from least to greatest.

1) Order the expressions from least to greatest: √49, √643

, √753

Objective #17: Can you estimate cube roots?

a) The value of √553

is between which two integers?

A. 2 and 3

B. 3 and 4

C. 4 and 5

D. 6 and 7

a) Which answer shows the expressions ordered from least to greatest?

√643

, 2, √273

, √1003

A. √643

, 2, √273

, √1003

B. 2, √643

, √1003

, √273

C. 2, √273

, √643

, √1003

D. √1003

, √643

, √273

, 2

Simplifying Radicals: Make sure to use the correct order of operations!

1) Parenthesis

2) Exponents

3) Add/Subtract

4) Square Root

Example:

1) √(3 − 1)2 + (2 + 1)2 2) √(5 − 7)2 + (1 + 1)2

Objective #18: Can you simplify expressions with exponents and

radicals?

a) √(6 − 5)2 + (3 + 1)2 b) √(2 − 8)2 + (5 + 3)2