big ideas, and new vocabulary

1

Math 1201

Unit 4: Roots & Powers

Read Building On, Big Ideas, and New Vocabulary, p. 202 text.

Ch. 4 Notes

§4.1 Estimating Roots (1 class)

Read Lesson Focus p. 4 text.

Outcomes

1. Define and give an example of a radical. pp. 204, 539

2. Identify the index and the radicand of a radical. p. 204

3. Determine the exact value, or an approximation of the exact value, of the root of a number. p.

205

nDef : A radical is an expression consisting of a radical sign (radical symbol), a radicand, and an

index.

E.g.: 3 52 4 2 327

9 9; 25; 100; 110; 0.25 0.25;81

E.g.: Complete the table below.

General Form of Radical Example Index Radicand 2x x (Square Root) 29 9 2 9

2x x 20.25 0.25 0.25

2x x 2

36 36

81 81 2

3 x (Cube Root) 3 27 3 27

3 x 3 0.001 3

2

3 x 3

64

125

64

125

4 x (Fourth Root) 4 126 126

4 x 4 12.58 4

4 x 4

16

625

16

625

5 x (Fifth Root) 5 18

5 x 5 0.00056

5 x 5

22

13

Evaluating Radicals

Evaluating a radical means to write it as an exact or as an approximate value.

Complete the following table.

Radical Value Exact? or

Approximate?

Why that Value?

Because … 2 9 or 9 3 Exact 23 9

2 18 or 18 4.242640687… Approximate 2

4.242640687 18

2 18 or 18 ???????? ????????

2 0.25 or 0.25 0.5 Exact 2

0.5 0.25

27 7

or 11 11

0.7977240352… Approximate 2 7

0.797724035211

21 1

or 9 9

1

or 0.33

Exact

21 1

3 9

3 16

3 27 Exact

3 0.64

3 0.64

316

81

4 16

4 16

4 27

416

81

4 0.64

5 32 Exact

3

5 32 5

2 32

53

8 Approximate

E.g.: Evaluate to two decimal places, if necessary and if possible. Explain why your answer is correct.

The first four are done for you.

a) 250

250 15.81 because 2

15.81 250

b) 121

121 11 because 2

11 121

c) 36

36 ??????????

d) 3 85

3 85 4.40 because 3

4.40 85

e) 3 1728

3 1728 _____________ because 3

___________ 1728

f) 3 915.0625

3 915.0625 _____________ because 3

___________ 915.0625

g) 3 0.000027

3 0.000027 _____________ because 3

___________ 0.000027

h) 4 92.3521

4 92.3521 _____________ because 4

___________ 92.3521

i) 4 625

4 625 ??????????

j) 5 243

4

5 243 _____________ because 5

___________ 243

k) 532

3125

532

_____________3125 because

5

32

_____________ 3125

Radicals with Negative Radicands

*******Negative radicands can only be evaluated if the INDEX is ODD.

Radicals with negative radicands cannot be evaluated if the index is EVEN. This is because no number

can be multiplied by itself an even number of times to give a negative answer.

E.g.: Determine if it is possible to evaluate each of the following radicals.

a) 300 Y OR N

b) 300 Y OR N

c) 3 300 Y OR N

d) 3 300 Y OR N

e) 4 300 Y OR N

f) 4 300 Y OR N

g) 5 300 Y OR N

h) 5 300 Y OR N

E.g.: Evaluate each of the following radicals rounded correctly to FOUR decimal places, if necessary

and if possible.

a) 3 300 _______________

b) 3 15.62590 _______________

c) 3 27000 _______________

d) 3 300 _______________

e) 38

_____________27

f) 3 15625 _______________

g) 3 16 _______________

h) 4 16 _______________

i) 4 16 _______________

j) 4 16 _______________

5

k) 532

_______________3125

l) 5 3125 _______________

m) 6 3125 _______________

E.g.: For each given number, write an equivalent form as a radical.

Number Equivalent Square

Root

Equivalent Cube

Root

Equivalent Fourth

Root

Equivalent Fifth Root

5 25 3 125 5 3125

0.8 0.64 3 0.512 4 0.4096

-7 Not possible Not possible 5 16807

2

7

3

8

343 4

16

2401 5

32

16807

-0.25 Not possible 3 0.015625 5 0.00009765625

E.g.: Choose values for x and n so that n x is:

a) A whole number 0,1,2,3,4,5,6,

Possible Answers: 3 5425 5; 27 3; 625 5; 32768 8

b) A negative integer 4, 3, 2, 1

Possible Answers: 3 527 3; 32768 8

c) A rational number (a fraction, a decimal that ends, or a decimal that repeats)

Possible Answers: 3 4121 11 1

; 76.765625 4.25; 0.164 8 6561

d) An irrational number (an approximate decimal)

Possible Answers:

5432

27 5.196152423 ; 0.8735804647 ; 1.331335364 ; 33 2.0123466173

Do #’s 1, 2, 4 c, 5 a, d, e, f, 6, p. 206 text in your homework booklet.

6

§4.2 The Real Number System (1 class)


Outcomes

1. Explain what is meant by classifying numbers. See notes

2. Classify real numbers. p. 209

3. Define and give examples of irrational numbers. pp. 208, 537

4. Give the symbol used to represent the irrational numbers. See notes

5. Order irrational numbers. pp. 209-210

nDef : Classifying numbers means determining to what set(s) of numbers a given number belongs. In

order to classify numbers, you need to know the different sets of numbers.

Sets of Numbers

1. Natural Numbers : The counting numbers. 1,2,3,4,5,6,7,

2. Whole Numbers : The counting numbers plus zero. 0,1,2,3,4,5,6,7,

3. Integers or : The whole numbers plus the opposites of the natural numbers.

or 7, 6, 5, 4, 3, 2, 1,0,1,2,3,4,5,6,7,

4. Rational Numbers : Numbers that can be written as fractions OR decimals that repeat OR

decimals that end. E.g.: 3

, 4.75, 1.37

. , , , 0

aa b b

b

5. Irrational Numbers or : Numbers that CANNOT be written as fractions OR decimals

that end OR decimals that repeat. E.g.: 3 4 52

2, 5.764, 71, , ,3

e .

, , , 0a

a b bb

6. Real Numbers : The rational numbers and the irrational numbers combined.

7. Complex Numbers : Numbers that can be written in the form a bi where ,a b and

1i . E.g.: 4 42, 15, 3 4 ,

9i i

These numbers can be represents in a Venn diagram (see below).

7

Important observations you need to make from the chart.

Observation #1:

Notice that 9 is a natural number. It is because 9 3 .

Observation #2:

Notice that the only difference between natural numbers and whole numbers is the zero.

Whole numbers = Natural numbers + zero

Observation #3:

Notice that the difference between whole numbers and integers are the negative numbers.

Integers = Whole numbers + the negatives of the whole numbers

8

Observation #4:

All integers are fractions. Not all fractions are integers.

E.g.: -2 is an integer and can be written as 2 2 2

or or 1 1 1

to make it a fraction.

However, 1

0.33333333333333 is not an integer.

Observation #5:

Fractions can be written as a terminating decimal or a repeating decimal

E.g.: 1

0.52 and 0.5 is a terminating decimal.

10.3333333333333 0.3

3 is a repeating decimal.

Observation #6:

Rational numbers = Integers + fractions

Observation #7:

Irrational numbers are numbers that cannot be written as a fraction.

E.g.: , 7

Another way to see them is that they are neither repeating decimals nor terminating decimals.

Observation #8:

Real numbers = rational numbers + irrational numbers. Every number you know of is a real number.

Observation #9:

The difference between complex numbers and real numbers is that complex numbers give solutions for

the following expressions and more! Every number is a complex number.

7, 1 9, 25 5i

Complete the following table. The first one and the last one are done for you.

Number W I or

4.5

121

2

5

9

0

0.313233…

3

-6

Ordering Irrational Numbers

Change each number to a decimal approximation.

E.g.: Order 3 3 42, 2, 6, 11, 40 from least to greatest.

3

3

4

2 1.4142

2 1.2599

6 1.8171

11 3.3166

40 2.5149

So the order from least to greatest is 3 3 42, 2, 6, 40, 11 .

Do #’s 3, 4, 7, 11, 14, 15, p. 211 text in your homework booklet.

10

§4.3 Mixed and Entire Radicals (2 classes)


Outcomes

1. Simplify radicals. pp. 215-216

2. Define and give an example of an entire radical. pp. 217, 536

3. Define and give an example of a mixed radical. pp. 217, 538

4. Rewrite an entire radical as a mixed radical. p. 214, 216

5. Rewrite a mixed radical as an entire radical. p. 214, 217

To help simplify radicals, we need to use a special property of radicals and some special numbers (i.e.

perfect squares, perfect cubes, and so on).

Multiplication Property of Radicals

Determine if the following expressions are equal.

Expression 1 Expression 2 Expression 3 Equal? Y or N

12 4 3 4 3 Y

99 9 11 9 11

32 16 2 16 2

3 24 3 8 3 3 38 3

3 54 3 27 2 33 27 2

3 256 3 64 4 33 64 4

4 32 4 16 2 44 16 2

4 243 4 81 3 4 481 3

4 512 4 256 2 44 256 2

All the examples in the table above illustrate the Multiplication Property of Radicals.

***** n n na b a b

Note that this property works both ways:

i. n n na b a b 33 354 27 2 and

ii. n n na b a b 33 327 2 54

Simplifying Radicals

To simplify radicals, we will use the Multiplication Property of Radicals and some special numbers.

11

Special Numbers used to Simplify Radicals

Complete the table below. Some have been done for you.

Perfect Squares

(Used to simplify Square

Roots)

Perfect Cubes

(Used to simplify Cube

Roots)

Perfect Fourths

(Used to simplify

Fourth Roots)

Perfect Fifths

(Used to simplify Fifth

Roots) 21 1 31 1 41 1 51 1 22 4 32 8 42 16 52 32 23 9 33 27 43 81

53 243

310 1000 410 10000 510 100 000

220 400

We are going to use the Multiplication Property of Radicals and the special numbers in the table to

simplify radicals. The special numbers that we use depend on whether we are simplifying a square root,

a cube root, a fourth root, or a fifth root.

E.g.: Simplify 99 .

Since we are dealing with a square root, we look in our table for the BIGGEST perfect square that

divides into 99. That perfect square is 9. Using the Multiplication Property of Radicals gives

99 9 11 3 11

E.g.: Simplify 3 54 .

Since we are dealing with a cube root, we look in our table for the BIGGEST perfect cube that divides

into 54. That perfect cube is 27. Using the Multiplication Property of Radicals gives

3 33 354 27 2 3 2

12


Since we are dealing with a fourth root, we look in our table for the BIGGEST perfect fourth that

divides into 48. That perfect fourth is 16. Using the Multiplication Property of Radicals gives

4 4 4 448 16 3 2 3


Since we are dealing with a fifth root, we look in our table for the BIGGEST perfect fifth that divides

into 96. That perfect fifth is 32. Using the Multiplication Property of Radicals gives

5 5 5 596 32 3 2 3

How do you know when to stop? You stop when the no number in the column you are using divides

evenly into the radicand.

nDef : An entire radical is a radical of the form n x .

E.g.: 99 , 3 54 , 4 48 , 5 96

nDef : A mixed radical is a radical of the form na x , where 1a .

E.g.: 3 11 , 33 2 , 42 3 , 52 3

E.g.: Write 48 as a mixed radical.

48 16 3 16 3 4 3

E.g.: Write 4 3 as an entire radical. Since 16 4 , we can write

4 3 16 3 16 3 48

E.g.: Write 3 54 as a mixed radical.

3 33 3 354 27 2 27 2 3 2

E.g.: Write 33 2 as an entire radical. Since 3 27 3 , we can write

3 33 3 33 2 27 2 27 2 54


4 4 4 4 448 16 3 16 3 2 3


13

4 4 4 4 42 3 16 3 16 3 48


5 5 5 5 596 32 3 32 3 2 3


5 5 5 5 52 3 32 3 32 3 96


Entire Radical n a b n na b Mixed Radical

75 25 3 25 3 5 3

45

72

3 16 3 8 2 33 8 2 32 2

3 81

3 250

4 160 4 16 10 4 416 10 42 10

4 80

4 512


Mixed Radical n na b n a b Entire Radical

6 3 36 3 36 3 108

4 5

7 10

32 5 3 38 5 3 8 5 3 40 33 4

35 7

43 4 44 81 4 4 81 4 4 324

43 8

45 5

Do #’s 4, 5, 9, 10 a,c,e,h, 11 a,c,e,g,i, 12 a,c,e,g,i, 14, 17 a,c, 18 a,c, 20, 21, pp. 218-219 text in your

homework booklet.

14

§4.4 Fractional Exponents and Radicals (2 classes)


Outcomes

1. Evaluate radicals with fractional exponents. pp. 223-226

2. Explain how 1

and n nx x are related. p. 223

3. Explain how and m

mn nx x are related. p. 225

4. Rewrite a radical as an equivalent power. p. 225

In this section you will convert radicals to an equivalent power, convert a power to an equivalent radical,

simplify powers which have exponents which are fractions, and extend some basic laws of exponents to

include exponents that are fractions.

The Relationship between the Radical n x and the Power

1

nx


Radical Power Radical Power Radical Power 2x x

1

2x Equal

Y or

N?

3 x 1

3x Equal

Y or

N?

4 x 1

4x Equal

Y or

N? 29 9

1

29 Y 3 64

1

364 Y 4 625

1

4625 Y

211 11 1

211 Y 3 12

1

312 Y 4 34

1

434 Y

245.678 45.678 1

245.678 Y 3 10.68

1

310.68 Y 4 1001.28

1

41001.28 Y

23 3

4 4

1

23

4

Y 3

11

4

1

311

4

Y 4

89

24

1

489

24

Y

3 8 1

38 Y Y

From the table above you should see that the radical n x is the same as the power

1

nx .

When n is a natural number (1, 2, 3, 4, …) and x is a rational number (decimal that ends or repeats), then

1

*************** n nx x

E.g.:

11 1 13

542 43 58 8 2

25 25 5; ; 5.0625 5.0625 1.5; 7776 7776 6125 125 5

15

E.g.: Complete the following table.

Want do you think the exponent 1

2 means? The exponent

1

2 means the square root.


3 means?


4 means?


5 means?


n means? The exponent

1

n means the n

th root.

Recall that fractions such as 1 1 1

, , and 2 4 5

can be written as the terminating decimals 0.5, 0.25, and 0.2

This may be useful in question # 4, p. 227 text.

E.g.: Complete the table below. The first one is done for you.

Radical n x Equivalent Power

1

nx

2217 217 1

2217 3 50

4 25

5 30

E.g.: Evaluate each without using a calculator:

11 1 1 43 2 3

811000 ; 0.25 ; 8 ; ;

16

1

331000 1000 10 1

20.25 0.25 0.5

1

338 8 2

1

44

81 81 3

16 16 2

Do #’s 3, 4 a, b, d, 5, 6, 13 a, c, e, 14 b, c, d, p. 227 text in your homework booklet.

16

The Relationship between the Radical m

n x and the Power

m

nx


Radical m

n x Power m

nx Equal Y or N?

3

4 16 3

416 Y

2

5 27 2

527 Y

4

32

5

4

32

5

Y

3

17.625 3

217.625 Y

From the table above you should see that the radical m

n x is the same as the power

m

nx .

When m and n are a natural numbers (1, 2, 3, 4, …) and x is a rational number (decimal that ends or

repeats), then

***************m

mn nx x

223 5

3 5342 43

225 5

8 8 4E.g.: 25 25 125; ; 5.0625 5.0625 7.59375

125 125 25

7776 7776 36

E.g.: Complete the table below.

Radical m

n x Equivalent Power

m

nx

4 32 1

432

3

4 5 0.755

2

526

5

315

3.56

11

0.7512

17

E.g.: Evaluate each without using a calculator:

33 3 34 32 0.44 2 43 2

8181 ; 0.01 ; 27 ; ; 32 ; 4 ; 16

16

3

334481 81 3 27

3 3 332

2 21 1 1 1

0.01 0.001100 100 10 1000

4 4 43327 27 3 81

3 3 3281 81 9 729

16 16 4 64

2 20.4 25532 32 32 2 4

3 3

24 4 impossible

3 3 3 344 416 1 16 1 16 1 2 1 8 8

E.g.: The value (V) of a car is given by the formula 232000 0.85t

V where t represents the age of the

car in years. Find the value of the car after 5 years.

If 5t then,

5

2.5232000 0.85 32000(0.85) $21315.59V

Do #’s 8, 10 a, b, c, e, 11, a, b, c, 12, 16 a, 17, 18, 19, 20 b, pp. 227-228 text in your homework

booklet.

18

§4.5 Negative Exponents and Reciprocals (2 classes)


Outcomes

1. Evaluate powers with negative exponents. pp. 231-232

2. Explain how 1

and n

nx

x

are related. p. 231

3. Explain how 1

and n

nx

x are related. p. 231

4. Explain how

n

p

q

and

n

q

p

are related pp. 231-232

In this section you will evaluate powers with negative exponents.

The Relationship between the Power nx and the Power 1

nx


nx 1

nx Equal Y or N?

25 2

1

5

3

4

3

1

4

43.25 4

1

3.25

52 5

1

2

From the table above you should see that the power nx is the same as the power 1

nx.


1************** n

nx

x

E.g.: 2 3 4 5

2 3 4 5

1 1 1 1 1 1 1 16 ; 5 ; 4 ; 3

6 36 5 125 4 256 3 243

Do #’s 3 a, c, 4, a, b, 5, 6, 9, a, b, c, d, 10, a, c, d, 13 a, b, c, p. 233 text in your homework booklet.

19

The Relationship between the Power nx and the Power

1nx


nx 1

nx Equal Y or N?

25 2

1

5 Y

3

4

3

1

4

Y

43.25 4

1

3.25 Y

52 5

1

2 Y

From the table above you should see that the power nx is the same as the power

1nx

.


1************** n

nx

x

E.g.: 2 3 4 5

2 3 4 5

1 1 1 16 36; 5 125; 4 256; 3 243

6 5 4 3

Do #’s 3 d, 8 f, p. 233 text in your homework booklet.

The Relationship between the Power

n

p

q

and the Power

n

q

p


Power

n

p

q

Power

n

q

p

Equal Y or N?

24

3

23

4

Y

31

6

36

1

Y

42

5

45

2

Y

57

3

53

7

Y

20

From the table above you should see that the power

n

p

q

is the same as the power

n

q

p

.

When n is a natural number (1, 2, 3, 4, …), p and q are integers with 0q then

**************

n n

p q

q p

E.g.:

2 2 3 3 4 4 5 51 5 2 7 343 3 5 625 4 7 16807

25; ; ;5 1 7 2 8 5 3 81 7 4 1024

Do #’s 3 b,7, 8 d, e, 12, 13 d, e, f, p. 233 text in your homework booklet.

21

§4.6 Applying the Laws of Exponents (2 classes)


Outcomes

1. Simplify expressions using the laws of exponents. pp. 238-241

In this section you will review the laws of exponents and use them to simplify expressions containing

variables.

Recall the following laws of exponents from previous math courses.

Exponent Law Exponent Law as an Equation Example

Product of Powers m n m na a a 2 3 2 3 56 6 6 6 7776

Quotient of Powers ; 0

mm n m n

n

aa a a a

a

3

3 2 3 2 1

2

66 6 6 6 6

6

Product of a Power n

m m na a 3

2 2 3 66 6 6 46656

Power of a Product 2n n na b ab a b

4 4 43 5 3 5 50625

Power of a Quotient n n

n

a a

b b

2 2

2

7 7 49

4 4 16

Complete the table below. The first one is done for you.

Product of Powers m n m na a a 2 3 2 3 54 4 4 4 1024 m n m na a a 21.3 1.3 m n m na a a 2 7

1 1

2 2

m n m na a a 7 9y y

22


Quotient of Powers m

m n

n

aa

a

6

6 4 6 4 2

4

88 8 8 8 64

8

mm n

n

aa

a

8

8 5

5

1.31.3 1.3

1.3

mm n

n

aa

a

10

10 5

5

1

1 12

2 21

2

mm n

n

aa

a

7

7 9

9

yy y

y


Product of a Power

n

m m na a 2

4 4 2 88 8 8 16777216

n

m m na a 10

21.3

n

m m na a 3

53

4

n

m m na a 8

11y


Power of a Product

n n n na b ab a b

2 2 28 3 8 3 64 9 576

n n n na b ab a b

101.3 0.8

n n n na b ab a b

32 16

7 5

n n n na b ab a b

8r s

23


Power of a Quotient n n

n

a a

b b

3 3

3

8 8 512 or 18.962

3 3 27

n n

n

a a

b b

51.3

0.8

n n

n

a a

b b

23

45

8

n n

n

a a

b b

8r

s

Now let’s use these laws of exponents to simplify expressions which have negative and/or fractional

exponents.

E.g.: Write 2 70.8 0.8 as a single power.

2 72 7 5

5

10.8 0.8 0.8 0.8

0.8

E.g.: Write

3 52 4

4 4

5 5

as a single power.

3 52 4 2 3 4 5 6 20 6 20 14

4 4 4 4 4 4 4 4

5 5 5 5 5 5 5 5

E.g.: Write

53

5

1.5

1.5

as a single power.

5

3 3 5 1515 5 10

5 5 5

1.5 1.5 1.51.5 1.5

1.5 1.5 1.5

24

E.g.: Write

5 1

4 4

3

4

9 9

9

as a single power.

5 15 1 4

4 3 14 44 4 44 4 4

3 3 3

4 4 4

9 9 9 99 9

9 9 9

E.g.: Simplify

22

33 3

4 4

2 2 6 8 8 82 83 3 3 3

38

327 27 27 27 27 3 6561

64 64 64 64 64 4 4 65536

E.g.: Simplify

32

3

t

t

32 6

6 3 9

3 3

t tt t

t t

E.g.: Simplify 4 2 2 3m n m n

2 34 2 2 3 4 2 2 3 4 2 6m n m n m m n n m n m n

E.g.: Simplify 4 3

2

6

14

x y

xy

4 3 4 3 3

3 24 1 3 5 3

2 2 5 5

6 6 3 3 3 1 3

14 14 7 7 7 7

x y x y xx y x y x

xy x y y y

E.g.: Simplify 3

4 2 225a b

3 4 3 2 3 12 63 3 3 3

4 2 4 2 3 6 32 1 2 1 2 2 22 2 225 25 25 5 125a b a b a b a b a b

25

E.g.: Simplify 3 1

3 12 2x y x y

3 13 1 3 1 2 2

3 13 1 3 1 2 2 1 22 22 2 2 2 21

1 xx y x y x x y y x y x y x y x

y y

E.g.: Simplify

55 2

1 1

2 2

12

3

x y

x y

5 5

5 11 10 1 5 1 11 6 11 65 5 32 2 532 22 2 2 2 2 2 2 2 2

1 1 1 1 11 11

2 2 2 2 2 2

12 12 1 44 4 4 4 4

33

x y x y yx y x y x y x y y

x y x y x x

E.g.: Simplify

12 4 2

4 7

50

2

x y

x y

11 1 1 112 4 2 4 22 2 2 2

2 3 21 1 34 7 4 7 2 3 2 3

2 32 2 2

50 50 1 1 25 25 525 25

2 2

x y x yx y

x y x y x y x yx y xy

E.g.: Simplify

2

3

2 3

6

x xy

xy

2 2 2 2 2 2 3 2 3 2

3 1 2 3

3 3 3 3 3

22 1 2

2 3 2 3 2 9 18 183

6 6 6 6 6

1 33 3

x xy x x y x x y x y x yx y

xy xy xy xy x y

xx y x

y y

E.g.: Simplify

11 2 1 1

5 4 4 2x y x y

1 1 1

1 1 11 1 12 1 1 2 1 1 1 1 12 2 25 5 104 4 2 4 4 2 4 4 2

71 1 1 21 1 1 2 5 7 71 1 1 3 204 2 4 410 10 4 20 20 20 204 4 2 4

3 3

4 4

1

x y x y x y x y x y x y

xx x y y x y x y x y x

y y

26

E.g.: A cone with equal height and radius has a volume of 318cm . What are the radius and the height to

the nearest tenth of a centimetre?

2

3

r hV

Since the volume is 18 and r h then

2

3

3

3

183

183

3 18 33

54

54

r r

r

r

r

3r

3

3

54

54

2.6cm

r

r

r

The radius and height are about 2.6cm.

Do #’s 3 a, b, d, 4, 5, 6, 7, 8 a c, e, g, 9 a, c, e, g, 10 b, d, f, h, 11, 12, 14, 15 c, d, 16, 17, 19, pp. 241-

243 text in your homework booklet.

Do #’s 1 b, d, 4 a, c, 6 a, c, d, e, f, h, i 9, 10, 11, 12, 14, 17 b, d, 18 b, d, 19, 20, 24, 28 b, c, 29 b, d, 30

32, pp. 246-248 text in your homework booklet.

big ideas, and new vocabulary

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