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NEL46

MF9SB_CH02_p46-69 pp7.qxd 4/7/09 3:51 AM Page 46

GOALS

You will be able to

• represent repeated multiplication usingpowers

• simplify expressions involving powers

• solve problems involving powers

• communicate about calculationsinvolving powers

• calculate and estimate square rootsof positive rational numbers

2Chapter

Powers,Exponents, and

Square Roots

NEL 47

How many small cubes are in eachlarge cube?


CHAPTER 2 Getting Started

Product DisplayNicole works part time at an electronics store. She is setting up a displayof video game consoles. Each console is in a cube-shaped box with a sidelength of 60 cm. The display must be in the shape of a cube.

NEL48 Chapter 2 Powers, Exponents, and Square Roots

YOU WILL NEED

• centimetre cubes

What could be the volume of the display?

A. Arrange some centimetre cubes to make a larger cube.

B. Suppose each cube represents one box in the display. What are thedimensions of your display?

C. How can you calculate the floor area of your display?

D. How can you calculate the volume of your display?

E. What is the volume of your display?

F. Repeat parts A to E using two other arrangements of centimetre cubes.

?


NEL 49Getting Started

WHAT DO You Think?Decide whether you agree or disagree with each statement. Explain yourdecision.

1. A square can have an area of

2. A cube can have a side length, area of a face, and volume with thesame numerical value.

3. There is one method you can use to factor a number.

4. When you square a number, the result is just as likely to be less thanthe original number as it is greater.

8.41 cm2.

x cm

area � x cm2

volume � x cm3

1 2 4 1684

MF9SB_CH02_p46-69 pp7.qxd 4/7/09 3:51 AM 9

Modelling Squares and Cubes2.1

Represent perfect squares and perfect cubes using models.

GOAL

INVESTIGATE the Math

YOU WILL NEED

• a calculator• centimetre cubes• centimetre grid paper


power

a numerical expression thatshows repeated multiplication;e.g., the power is a shorterway of writing It isread as “two to the third” or“two cubed“—2 is the baseand 3 is the exponent. Wesay 2 has the exponent 3.

2 3 2 3 2.23

base

the number used as a factor ina power

exponent

the number used to expressthe number of factors in apower

Yvonne is making a gift for hersister’s Naming Ceremony. It willbe a cube with a square photo oneach face. The sides of the photoswill be natural number centimetrelengths. She wants the cube to beas large as possible, but she ismailing it and it cannot be morethan in volume.5000 cm3base

exponent23

Side Length of Side Length Area of Face of Area of Face Volume of Cube, Volume of CubeCube, s (cm) as a Power, Cube, (cm2) as a Power, (cm3) as a Power,

2

4

8 8 3 8 5 64

434 3 4 3 4 5 64

232 3 2 3 2 5 8222 3 2 5 421

s3s 3 s 3 ss2s 3 ss1

What should be the dimensions of the cube?

A. Complete the second and third rows in the table expressing eachside length, face area, and volume as a power.

?

B. Continue to complete rows for other side lengths as necessary.

C. What should be the side length of Yvonne’s cube? Explain how you know.

Reflecting

D. You can represent the value 64 with two different models, and How is the model of the form different from the model of theform ?

E. How do you know that 225 can represent the area of one of thesquare faces of a cube with natural number centimetre lengths, butnot the volume?

F. How do you know that 343 can represent the volume of a cube likeYvonne’s, but not the area of one face?

s3s2

s3.s2

MF9SB_CH02_p46-69 pp7.qxd 4/7/09 3:51 AM 0

WORK WITH the Math

NEL 512.1 Modelling Squares and Cubes

perfect square

the product of a naturalnumber multiplied by itself;e.g., 49 is a perfect squarebecause 7 3 7 5 49.

EXAMPLE 1 Modelling square powers

A square wall tile has an area of .Represent the area of this tile as a geometricmodel and as a power.

Bay’s Solution

100 cm2

1728 is a perfect cube, so I wrote it as apower using a base of 12 and an exponentof 3.

1728 5 123

Each side of the box must have the samelength and the base must be a square. Iknew the cube would be more than 10 cm on a side, because 1728 is more than103 5 1000.

The box is a cube.

EXAMPLE 2 Modelling cube powers

I tried 12. Since the volume is even, I knewthe side length must also be even.

12 3 12 3 12 5 1728

The geometric model is a cube that looks like this.

12 cm

12 cm12 cm

A softball comes in a cube-shaped box with a volume of . Represent the volume of this box as a geometric model and as a power.

Amanda’s Solution

1728 cm3

perfect cube

the product of a naturalnumber multiplied by itselftwice; e.g., 343 is a perfectcube because7 3 7 3 7 5 343.

The tile is square, so the geometricmodel must be square too. Since

each side of thesquare must be 10 cm.100 5 10 3 10,

10 cm

10 cm

100 is a perfect square, so Iwrote it as a power using a baseof 10 and an exponent of 2.

100 5 102


Checking

1. Represent each geometric model as a power.a) b)

3

332

2


In Summary

Key Idea

• You can represent some powers usinga geometric model. For example, youcan represent a perfect square as thearea of a square with natural-number-length sides and a perfect cube as thevolume of a cube with natural-number-length sides.

Need to Know

• A perfect square can be written as a power: • A perfect cube can be written as a power: n3 5 n 3 n 3 n.

n2 5 n 3 n.

xx x

x2xxx3

two dimensions

three dimensions

2. a) Write as a power. b) Write as a power.

3. a) Determine the side length of a square with an area of b) Determine the side length of a cube with a volume of

Practising

4. Determine the value of .a) c) e)b) d) f )

5. A square floor mat has a side length of 5 m. Write the area of themat as a power.

6. The side length of a cube is 12 cm. Determine the following:a) the area of one face b) the surface area c) the volume

7. Joga is making palak paneer. He used a large cube of cheese thathad a volume of a) Sketch a model of the cheese. Label the side lengths.b) Joga sliced the cheese into 3 cm cubes. How many cubes did

he have?

3375 cm3.

53 5 ■■3 5 273 3 3 3 3 5 ■372 5 ■■2 5 1004 3 4 5 ■2

■

8 cm3.81 m2.

11 3 116 3 6 3 6


NEL 532.1 Modelling Squares and Cubes

8. How many more perfect squares than perfect cubes are therebetween 1 and 1000?

9. Multiple choice. A square floor mat has a side length of 22 m.What is the area of the mat as a power?A. B. C. D.

10. Multiple choice. Determine the area of one face of a cube with aside length of 14 cm.A. B. C. D.

11. Multiple choice. Determine the volume of a cube with a sidelength of 14 cm.A. B. C. D.

12. Sketch geometric models for and . How are the models alikeand different?

13. Austin says that he can draw a geometric model for any power of 2.Do you agree or disagree with him? Justify your decision.

14. Two perfect squares have a difference of 169.a) How far apart are the square roots?b) How far apart are the cubes of the values in part a)?

15. Which numbers have the same values as their square and their cube?

16. Nasri is creating a mosaic using tiles for art class. He has a frame that is60 cm by 60 cm and divided into four sections. The frame’s border is2 cm wide. He has many tiles with these dimensions: 1 cm by 1 cm,2 cm by 2 cm, 3 cm by 3 cm, 5 cm by 5 cm, and 10 cm by 10 cm.Sketch some designs for Nasri’s mosaic. Use graph paper to help you.

Closing

17. How could you prove to someone that there are more perfectsquares than perfect cubes in the numbers between 100 and 200?

Extending

18. Nicole and her friend Hélène are preparing sucre à la crème. Theyuse plates that are 20 cm by 30 cm and cut the treats into 2 cmcubes. They will sell 10 cubes for $1.00. They hope to raise about$50. How many plates will Nicole and Hélène need?

19. Sean and Damien bought Patrick an MP3 player for his birthday.They have a sheet of wrapping paper that is 30 cm by 60 cm. Canthey wrap the box without cutting the paper? Sketch how you know.

20. You have seen that 64 is a perfect square and a perfect cube.Determine two other numbers with this property.

4342

2744 cm314 cm2196 cm3196 cm2

2744 cm314 cm2196 cm3196 cm2

222 m2222 m2223 m3222 m2

MP3 PLAYER14.6 cm


Expressing a Number as a Power2.2


Use powers to represent repeated multiplication.

GOAL

LEARN ABOUT the MathYvonne uses square sticky notes to leave messages for her mom. Shedecides to make a cube-shaped holder for the notes in woodworkingclass. She wants the holder to hold eight packs of notes. Each sticky-note pack is a cube and each sticky note is 8 cm wide.

YOU WILL NEED

• a calculator

What should the capacity of the container be??

Each sticky-note pack is a perfect cube; each pack hasdimensions 8 cm by 8 cm by 8 cm.

8 cm8 cm

8 cm

EXAMPLE 1 Representing volume using a power

Yvonne’s Solution

I want the holder to be in the shape of a cube and Iwant it to hold eight packs. I must make the large cubetwo packs wide, two packs high, and two packs long.The larger cube is 16 cm by 16 cm by 16 cm.16 cm

16 cm

16 cm

I described the capacity of my holder with a power.The side length is the base of the power and thenumber of dimensions is the exponent.

The capacity is

The capacity of the container is 163 cm3.16 3 16 3 16 5 4096 cm3.

Reflecting

A. How can Yvonne use the fact that to calculate ?

B. Why can you use powers to describe but not todescribe ?2 3 3 3 4

4 3 4 3 4,

16343 5 64


NEL 552.2 Expressing a Number as a Power

A base without an exponentis understood to have anexponent of 1; so, 5 5 51.

Communication Tip

I used a pattern. I started with an exponent of 3. Idecreased the exponent by 1. I noticed that whenthe exponent decreases by 1, the value of thepower is divided by 3.

I expect that 30 5 31 4 3 5 1.(3 4 3) 5 130 (9 4 3) 5 331

(27 4 3)5 3 3 3 5 932 33 5 3 3 3 3 3 5 27

EXAMPLE 3 Evaluating a power by using a pattern

Evaluate and

Nicole’s Solution

00.(21)0,50,30,

WORK WITH the Math

I wrote the power as a repeatedmultiplication. I had to repeateverything inside the brackets.

Q212R

25 Q21

2R 3 Q212R

I multiplied from left to right.2Q12R2

514

EXAMPLE 2 Evaluating a power

Evaluate and .

Derek’s Solution

2Q12R2Q21

2R2

I wrote the power as a repeatedmultiplication. I didn’t repeat the

minus sign with each becauseit wasn’t in brackets.

I multiplied from left to right.

12,

2Q12R

25 2

14

5 Q212R 3 Q12R

5 (21) 3 Q12R 3 Q12R 2Q12R

25 (21) Q12R

2

The two powers represent different numbers.

(5 4 5)5 150(25 4 5)5 551

(125 4 5)5 5 3 5 5 2552 53 5 5 3 5 3 5 5 125 I did the same with The result was the same.50.



I got the same result with although in thiscase, the value of the power just flipped between

and 1.21

(21)0,

5 1 (21 4 21)(21)05 21 (1 4 21)(21)1

5 (21) (21) 5 1 (21 4 21)(21)2 (21)3 5 (21) (21) (21) 5 21

I tried the same pattern with but it didn’t workthis time. I can’t write these expressions as theprevious value divided by 0, because division by 0is undefined, not 0. I can’t use the pattern todetermine the value of 00.

00

is undefined.00(0 4 0)5 0 3 0 5 ?02

03 5 0 3 0 3 0 5 0

In Summary

Key Idea

• Powers are used to represent repeated multiplication. The baserepresents the number being multiplied and the exponent, when it is awhole number, tells how many times the base appears. For example,

and

Need to Know

• Any power with a nonzero base and an exponent of 0 is equal to 1; thatis,

• If there are no brackets in a power, the exponent applies only to itspositive base: .

is the opposite of just as is the opposite of 3.• A power has a negative base when the base is negative and is enclosed

in brackets. For example, (23)4 5 (23) (23) (23) (23) 5 81.

2334,234

234 5 (21) (3 3 3 3 3 3 3) 5 281

x 2 0.x0 5 1,

Q67R3

5 Q67RQ67RQ

67R76 5 7 3 7 3 7 3 7 3 7 3 7

Checking

1. Represent each repeated multiplication as a power.

a) d)

b) e)

c) f ) Q34RQ34RQ

34RQ

34RQ

34R(24) (24) (24)

Q57RQ57RQ

57R(3.2 3 3.2) 3 (3.2 3 3.2)

2(7) (7) (7) (7) (7)5 3 5 3 5 3 5 3 5 3 5

Practising

2. Represent each repeated multiplication as a power.a) d)

b) e)

c) f ) Q23RQ23RQ

23RQ

23R(25.4) (25.4) (25.4)

Q89RQ89RQ

89RQ

89R(6 3 6) 3 (6 3 6)

2(8) (8) (8) (8) (8)4 3 4 3 4 3 4 3 4 3 4


NEL 572.2 Expressing a Number as a Power

Repeated Value in Power Base Exponent Multiplication Standard Form

6561

5

5

246

2(6) (6) (6)

22

(5) (5) (5)

94

3. Represent each power using repeated multiplication.a) b) c) d)

4. Evaluate each power.a) c) e)b) d) f )

5. Complete the table.

a)

b)

c)

d)

e)

6. Multiple choice. Which power does not represent 256?A. B. C. D.

7. Multiple choice. Which statement is true?A. C.B. D.

8. Multiple choice. Evaluate A. B. 25 C. 625 D.

9. Shelby says that for any power with a positive integer base, when thebase and exponent are switched, the greater power is always the onewith the greater base. Do you agree or disagree? Justify your decision.

10. If a power has a negative integer base, can you predict whether thepower has a positive or negative value? Explain.

11. Arrange in order from least to greatest.

12. Ihor read that, in Japan, some farmers grow watermelons insidecubes so the melons grow in the shape of a cube. He bought a sheetof special plastic that is 45.0 cm by 70.0 cm.a) Determine the area of the sheet of plastic.b) The surface area of a cube is where is the length of one

side. Determine the dimensions of the side length of the largestplastic cube Ihor can build.

s6s2,

(21)31(21)100,2(222),(22)4,224,

2542625(25)4.

262 5 36(21)6 5 21233 5 (3) (3) (3)3.13 5 3.1 3 3.1 3 3.1

162834428

(212)2234(27)3212.42(23)4273

2(22)4224(22)424



13. a) Calculate and b) Calculate and c) The fourth power of one number is 13 greater than the fifth

power of another. What are the numbers?d) How could you have predicted that the bases in part c) would

be fairly small?

14. a) Complete the pattern:

b) Use the pattern to evaluate where x is any whole number.

15. a) Order the following from least to greatest: .b) Would the order change if you replaced 6 with 5, or 0?

16. Order the following from least to greatest: .

17. Represent each repeated multiplication as a power.a) c)b) d)

18. Marilyn has 49 pennies, 32 nickels, 9 dimes, 25 quarters, 8 loonies,and 16 toonies in a jar.a) Write a power to represent the number of each type of coin.b) Write an equation using the powers in part a) to represent the

total number of coins.

Closing

19. Derek says since a power with a higher exponent is alwaysgreater. Do you agree? Explain.

Extending

20. a) Evaluate and b) Express each answer in part a) as a power with a base of c) Look for a pattern. How could you get the power in part b) just

by looking at the question in part a)?

21. Sue wanted to invite all 128 families of the Grade 9 class at herschool to the Math Olympics, an evening of math games andcontests. She didn’t have time to call every family herself, so shedecided to call two families and ask each person she called to calltwo more families, and so on.a) Determine how many rounds of calls will be needed.b) Represent the number of families as a power.

(22).(22) (22)5.(22)2(22)6,(22)3(22)4,

45 . 54,

2(p) (p) (p)(2y) (2y)

(t 3 t) 3 (t 3 t)s 3 s 3 s 3 s

5335,43,34,32,23,

25,6160,62,63,64,

1x,15 5 ■14 5 ■,

13 5 ■,12 5 1 3 1 5 ■,11 5 1,

55.45,35,25,54.44,34,24,


Math GAME

NEL 59Math Game

YOU WILL NEED• a deck of 40 cards

(no face cards)• a calculator

Super PowersNumber of players: 2 to 4

How to Play

1. On each turn, draw two cards from the deck.

2. Form a power with your cards, using one number as the base and one numberas the exponent.

3. Calculate the value of your power. Use a calculator to check your answer.

The value of the cards:

4. The player with the greater power on each turn wins one point.

5. Play until one player has 10 points.

Shelby’s Draw

I drew these two cards.

I could have chosen as a power butI chose instead because it was greater.

Yvonne’s Draw

I drew these two cards.

I chose

My power is greater, so I get the point.5 279 936

67 5 6 3 6 3 6 3 6 3 6 3 6 3 667.

3 3 3 3 3 5 19 68339 5 3 3 3 3 3 3 3 3 3 3 3 3

3993

1 2 3 4 5 6 78 910


Expressing a Numberin Many Ways

2.3


Represent a number in many ways using powers.

GOAL

EXPLORE the MathAmanda and Yvonne are playing a game. They have five numbers andthey want to see who can write a number the most ways using the sums,differences, products, or quotients of powers. The only rule is that theycannot use powers with an exponent of 0 or 1.

Amanda predicts you can write a greater number in more ways than alesser number. Yvonne doesn’t agree.

YOU WILL NEED

• a calculator

How could you decide whether Amanda is right or wrong??

54 5 52 3 32 2 33 2 122 54 5 63 4 22 54 5 92 2 33 54 5 52 1 52 1 22


Multiplying and Dividing Powers2.4

NEL 612.4 Multiplying and Dividing Powers

Simplify products and quotients of powers with the same base.

GOAL

INVESTIGATE the MathDerek wants to determine the value of this expression: He wonders if he can write it so that it will be easier to calculate the value.

How can Derek simplify this expression?

A. Rewrite Derek’s expression as a fraction, with powers in both thenumerator and denominator.

B. Write the numerator using repeated multiplication.

C. Express the numerator as a single power. How does this new powerrelate to the original powers in the numerator?

D. Write the denominator using repeated multiplication.

E. Express the denominator as a single power. How does this newpower relate to the original powers in the denominator?

F. Write the quotient as a single power. How does this new powerrelate to the other two?

G. How could Derek have simplified his original expression?

Reflecting

H. Why does it make sense that sometimes you add, sometimes yousubtract, and sometimes you multiply exponents to simplifyexpressions involving powers?

I. In each case, write a rule you can use to simplify• the product of two powers with the same base• the quotient of one power and another with the same base• a power raised to an exponent

J. Why do the bases need to be the same for some of the exponentrules you wrote in part I to work?

?

(54) (54) 4 (52)3.

YOU WILL NEED

• a calculator


The powers are to be multiplied, and theirbases are the same. I added the exponents.

a)5 36

(32) (34) 5 3214

The powers are to be divided, and their basesare the same. I subtracted the exponents.

b)5 62

65 4 63 5 6523

The power is to be raised to an exponent. I multiplied the exponents.

c)5 410

(42)5 5 4235

EXAMPLE 1 Simplifying numerical expressions using exponent laws

Simplify. a) b) c)

Bay’s Solution

(42)565 4 63(32) (34)

The powers are to be multiplied, and theirbases are the same. I added the exponents.

a)5 x11

(x6) (x5) 5 x615

The powers are to be divided, and the basesare the same. I subtracted the exponents.

b)5 x5

x 7 4 x 2 5 x722

The power is to be raised to an exponent. I multiplied the exponents.

c)5 x20

(x5)4 5 x534

EXAMPLE 2 Simplifying algebraic expressions using exponent laws

Simplify. a) b) c)

Derek’s Solution

(x5)4x 7 4 x 2(x6) (x5)


WORK WITH the Math

I added the exponents of the powers thatwere multiplied.5 (22)10 4 3(22)243

(22)7(22)3 4 3(22)243 5 (22)713 4 3(22)243

EXAMPLE 3 Simplifying using several exponent laws

Simplify. a) b)

Shelby’s Solution

( y3)5

( y) ( y4)(22)7(22)3 4 3(22)243



I multiplied the exponents of the power in the divisor.5 (22)10 4 (22)6

5 (22)10 4 (22)233

I subtracted the exponent of the divisor.

(22)7(22)3 4 3(22)243 5 (22)4 or 16

5 (22)45 (22)1026

I multiplied the exponents of the power inthe numerator and added the exponents ofthe powers in the denominator.

b)

5y15

y5

( y3)5

( y) ( y4)5

y335

y114

I subtracted the exponent of the divisor.

5 y105 y1525

I drew a diagram of my plan. The number ofpeople called doubles with each round. So, people will be called in round 1, people inround 2, and people in round 3.23

22

21

In round 8, people will be called using my plan.28To represent the number of calls in round 8 asa power, I think the base should be 2 and theexponent should be the number of the round.

EXAMPLE 4 Representing a power as an equivalent power

Austin and Shelby want to spread the news about school picture day. Austin will call twopeople and ask each one to call two more people, and so on. Shelby will call four peopleand ask each one to call four more people, and so on. Shelby says, with her plan, the samenumber of people would be called on the fourth round of calls as on the eighth round ofcalls with Austin’s plan. Is Shelby right?

Austin’s Solution: Representing as a power with a base of 428

1stround

22nd

round2 � 23rd

round2 � 2 � 2

I drew a diagram of the first two rounds ofShelby’s plan. The number of people called ismultiplied by 4 with each round. So, peoplewill be called in round 1, people in round 2,and so on.

42

41

1stround

42nd

round4 � 4



In Summary

Key Ideas

• Exponent law for productsTo simplify the product of two powers with the same base, keep thebase the same and add the exponents.

for example,

• Exponent law for quotientsTo simplify the quotient of two powers with the same base, keep thebase the same and subtract the exponents.

for example,

• Exponent law for a power of a powerTo raise a power to an exponent, keep the base the same and multiplythe exponents. for example,

Need to Know

• The exponent laws only work when the powers have the same base; forexample, you can’t multiply using the exponent law for powers.(32) (52)

(42)3 5 4233 5 46.(am)n 5 amn;

(25) 4 (23) 5 25–3 5 22.(am) 4 (an) 5 am2n(a 2 0);

(22) (23) 5 2213 5 25.(am) (an) 5 am1n;

Checking

1. Simplify.a) c) e)

b) d) f )

2. Express each number as a power with a different base.a) 16 b) c)

3. Simplify.a) b) c) (m3)4a64a8(x4) (x6)

9443

(88) (83) 4 (82)2(82)(54)2

5872 3 75 3 7

(118)5

11926

25(92) (97)

In round 4, people will be called using Shelby’s plan.44 To compare the number of people calledunder my plan to Shelby’s plan, I paired the 2sand wrote each pair as which is equal to 4.I knew I had four 4s multiplied together.After round 8, people would be calledunder my plan.

44

22,

Shelby is right.

5 44 5 (22)4 5 (22) (22) (22) (22)

28 5 (2 3 2) 3 (2 3 2) 3 (2 3 2) 3 (2 3 2)



Practising

4. Express each as a power with a single exponent.

a) c) e)

b) d) f )

5. Evaluate.

a) c) e)

b) d) f )

6. Determine the exponent that makes each statement true.a) c)b) d)

7. Multiple choice. Which is not equivalent to A. C.B. 2187 D.

8. Multiple choice. For which exponent is true?A. 1 C. 3B. 2 D. 4

9. Multiple choice. For which exponent is true?A. 3 C. 5B. 4 D. 6

10. Use a numerical example to illustrate each exponent law.a)b)c)

11. Oksana solved the following question:

When she checked the answer with her calculator she got 4.Identify the mistake Oksana made.


a) b) c) d) (p5)3

p11(s2)3(s5)y 7

y 2(x3) (x2)

5 32 768 5 215

5 224

29

23 3 28

(23) (23)2 5224

(23) (26)

(am)n 5 amnam 4 an 5 am2n (a 2 0)

(am) (an) 5 am1n

43 5 2■

24 5 4■

(33) (32) (32)

37312(33) (34)?

274 5 3■66 5 216■

6252 5 25■26 5 4■

(26)9(26)9

3(26)343(26)3

3 (28)243

(28)5(21)4(21)7

3(2310) 423(23)842

(211)7

(211)5(22)3(22)2

(57) (55)

(54)2(52)(23)3 4 24(34)2(33)

(63)5 4 (62)4125

122(106) (107)



13. Determine if each solution is correct or incorrect. If a solution isincorrect, correct the error and solve.

a) b) c)

14. a) Simplify by first writing the powers as products.

b) Simplify using the exponent law for quotients.

c) Evaluate .

d) How does knowing the exponent laws for quotients help explainwhy ?

e) Discuss whether would have a similar meaning for any valueof a (except 0).

15. How do you know that if the two powers are whole numbers?

16. Write each power in a simplified form.a) as a power of 2 c) as a power of b) as a power of 3 d) as a power of

17. Simplify.a) d)

b) e)

c) f )

Closing

18. Explain why but

Extending

19. a) Is there a whole number for which ? Explain why orwhy or not.

b) Can you write as a single power? Explain why or whyor not.

c) Can you write as a power of 5? Explain why or why or not.

CHAPTER 2

5x 1 5y

54 3 1253

320 5 4■

35 3 432 128.35 3 34 5 39,

(b) (b5) (b4)

b53(y) (y2) 43(a2) (a2) (a2)

(m5)2

m8

(a2)2(x4) (x2)2

(25)(2125)8275(23)9646

10■2 8■

a0a0 5 1

35

35

35

35

35

35

5 216 8075 45 275 (27)55 415 33

5(27)20

(27)155410

495310

37

5(27)15(27)5

(27)9(27)75410

(44) (45)5

(36) (34)

37

C(27)5D 3(27)5

C(27)3 D 3(27)748 3 42

(44) (42)3(32)3(34)

37


Mid-Chapter Review

NEL 67Mid-Chapter Review

FREQUENTLY ASKED QuestionsQ: How can you model perfect squares and perfect cubes?

A: You can use drawings or concrete materials to model perfect squaresor perfect cubes. A perfect square has two equal natural numberfactors: the length and width of a square. A perfect cube has threeequal natural number factors: the length, width, and height.

perfect square perfect cube

Q: How can you use powers to represent numbers?

A: You can use powers to represent repeated multiplication. Forexample, you can represent by thepower The exponent, 6, tells how many times the base, 5,appears in the power. You can use powers to represent perfectsquares or perfect cubes. For example, the perfect square 36 can be represented by the power The perfect cube 27 can be represented by the power

Q: How can you use the exponent laws to simplifyexpressions?

A: You can use the exponent laws to simplify numerical and algebraicexpressions.

33.62.

56.5 3 5 3 5 3 5 3 5 3 5

5 535 52125 5 5 3 5 3 525 5 5 3 5

• See Lesson 2.1, Examples1 and 2.

• Try Mid-Chapter Reviewquestions 1, 2, and 3.

Study Aid

• See Lesson 2.2, Examples 1,2, and 3.

• Try Mid-Chapter Review questions 4, 5, 6, and 7.

Study Aid

• See Lesson 2.4, Examples1, 2, and 3.

• Try Mid-Chapter Review questions 8, 9, 10, and 11.

Study Aid

Exponent Law Exponent Law Exponent Law forfor Products for Quotients Power of a Power

Statement ofExponent Law

Example5 465 325 25

(42)3 5 423334 4 32 5 3422(22) (23) 5 2213

(a 2 0)(am)n 5 amnam 4 an 5 am2n(am) (an) 5 am1n


PracticeLesson 2.1

1. Sketch a model to represent the following. Label each side length.a) c)b) d)

2. Calculate each dimension.a) the side length of a square with an area of 196 b) the dimensions of a cube with side face of 16

3. List two perfect squares between 0 and 1000 that are also perfectcubes. Show your work.

Lesson 2.2

4. a) Represent using repeated multiplication.b) Represent with two different powers.

5. a) Represent 216 using repeated multiplication.b) Represent 216 as a power. Show your work.

6. Evaluate each power without a calculator. Show your work.a) d)b) e)

c) f )

7. Put the answers to question 6 in increasing order.

Lesson 2.4


a) c) e)

b) d) f )

9. Express 1024 as a combination of powers using addition,subtraction, multiplication, and division. Identify as manypossibilities as you can.

10. To win a prize in a contest, Rafi had to answer the followingskill-testing question.

Express as a power of 2:

What should his answer be?

11. Simplify.a) c)

b) d)(a4) (a5)

(a2)3n10 4 n7

(d 3)9(b4) (b2) (b)

■(8) (64) 4 162 5

(26)7(26)5

C(26)4 D2(26)2(23)3

24(34)2(34)

(53)5

(52)4(29)4

(29)2(104) (107)

Q223 R

4234

80 (25)32(22)313

2828

cm2cm2

11 3 11 3 1182153212

16 cm2


Reading Strategy

Evaluating

Share your answers toquestions 6 and 7 witha partner. Do youagree? Defend yourresponses.


Google This!In 1920, mathematician Edward Kasner asked his nine-year-old nephew MiltonSirotta what name he should give to the number “A googol,” came the boy’sreply, and the name stuck. How large do you think a googol is?

1. What does mean?

2. If you were to write out in long hand, how many zeros would there beafter the 1?

3. The number is called a googolplex. Describe what the number would look like.

4. Express in another way.

5. How long does it take you to write all the digits of one million ? Whatabout one billion ? Suppose you could keep writing zeros withouttaking a break. About how long would it take you to write out the wholenumber equivalent of a googol? What about a googolplex?

6. Suggest three numbers greater than a googolplex. Explain how much longereach number would take to write out.

7. Why might the founders of Google have chosen this name for their searchengine?

(109)

(106)

10googol

10googol10googol

10100

10100

10100.

Curious MATH

NEL 69Curious Math


Combining Powers2.5


Simplify products and quotients of powers with the sameexponent.

GOALYOU WILL NEED

• a calculator

Nicole’s Solution

I calculated the surface area and volume ofYvonne’s cube. I wrote the side length of 20 as

to make it easier to compare to my cube.4 3 5

5 8000 cm3 5 64 3 125 5 43 3 53 5 (4 3 4 3 4) 3 (5 3 5 3 5) 5 (4 3 5) 3 (4 3 5) 3 (4 3 5)

Volume 5 (4 3 5)3

5 2400 cm2 5 6 3 16 3 25 5 6 3 42 3 52 5 6 3 (4 3 4) 3 (5 3 5) 5 6 3 (4 3 5) 3 (4 3 5)

Surface area 5 6 3 (4 3 5)2

LEARN ABOUT the MathNicole and Yvonne made origami paper cubes for a math project.

4 cm

Nicole´s cube

Yvonne´s cube

20 cm

How will the volume and surface area of Yvonne’s cubecompare to those for Nicole’s cube?

?

5 64 cm3 5 43

Volume 5 length 3 width 3 height

5 96 cm2 5 6 3 4 3 4 (or 6 3 16)

5 6 3 42 Surface area 5 6 faces 3 area of one face I calculated the surface area and volume of

my cube.

EXAMPLE 1 Comparing the surface area and volume of cubes


NEL 712.5 Combining Powers

Reflecting

A. How could Nicole have predicted she could calculate the surfacearea of Yvonne’s cube by multiplying her own cube’s surface areaby 25?

B. How could Nicole have predicted that she could calculate the volumeof Yvonne’s cube by multiplying her own cube’s volume by 125?

C. How do Nicole’s calculations show why and?(4 3 5)3 5 43 3 53

(4 3 5)2 5 42 3 52

I knew that , so I could use theexponent law or I could write

That’s the same as I realized that I could just multiply the oldvolume of by That’s an easy multiplication.

23.73

2 3 2 3 2 3 7 3 7 3 7.(2 3 7)3 5 (2 3 7) 3 (2 3 7) 3 (2 3 7)

14 5 2 3 7

The volume of the new cube is

The volume of a cube with a side length of 14 cm is8 3 343 5 2744 cm3.

5 8 3 73 5 23 3 73

143 5 (2 3 7)3 14 5 2 3 7

143.343 5 73

EXAMPLE 2 Simplifying the base of a power

Yvonne calculated the volume of a cube with a side length of 7 cm as 343 cm3. How canshe use that calculation to figure out the volume of a cube with a side length of 14 cm?

Yvonne’s Solution

I wrote the ratio of the surface area ofYvonne’s cube to the surface area of mycube, and then simplified.The surface area of Yvonne’s cube is 25 times greater

than that of my cube.

2400

965

25

1

I wrote the ratio of the volume of Yvonne’scube to the volume of my cube, and thensimplified.The volume of Yvonne’s cube is 125 times greater than mine.

800064

5125

1

WORK WITH the Math



Evaluate

Shelby’s Solution

25 3 54.

I noticed that can be expressedas a power with a base of 2,where or 24.42 5 (22)2

42 (23 3 42)3 5 (23 3 24)3

EXAMPLE 4 Simplifying expressions involving powers

I simplified using the product law.

5 (27)3 5 (2314)3

I could simplify even furtherusing the power of a power law.

5 221

Simplify

Austin’s Solution

(23 3 42)3.

I wrote the expression usingrepeated multiplication.

3 5 3 5 3 5 3 55 2 3 2 3 2 3 2 3 2

25 3 54

I rearranged the 2s and 5sbecause and that’seasier to multiply by than 2s or 5s.

2 3 5 5 10,3 (2 3 5) 3 (2 3 5)

5 2 3 (2 3 5) 3 (2 3 5)

I multiplied the 2s by the 5s.5 2 3 10 3 10 3 10 3 10

I simplified using powers.

5 20 0005 2 3 10 0005 2 3 104

EXAMPLE 3 Evaluating powers with different bases



Checking

1. Express as a product or quotient of two powers.

a) b) c) d)

2. Write each expression as a power with a single base. Show your work.

a) b) c) d) Q52

5 R4

(42 3 162)4(32 3 9)32 3 4

Q33

72R2

(32 3 54)3Q23R5

(2 3 3)4

I figured out what theexpression meant by usingrepeated multiplicationand the rules formultiplying fractions.

I realized that I could havejust applied the power tothe numerator anddenominator separately.

5 (232)3

(43)3

5 232 3 232 3 232

43 3 43 3 43

Q232

43 R3

5(232)

(43)3

(232)

(43)3

(232)

(43)

EXAMPLE 5 Simplifying powers in fraction form

I simplified using theexponent law for apower of a power. 5 236

49

5 23233

4333

Simplify

Derek’s Solution

Q232

43 R3

.

In Summary

Key Idea

• An exponent can be applied to each term in a product or quotientinvolving powers.

That is, and

For example, and .

Need to Know

• Sometimes an expression is easier to evaluate if you simplify it first; forexample, is easier to evaluate when it is simplified to

and is easier to evaluate if you rewrite it as asingle power of 8: 23 3 82 5 81 3 82 5 83.

23 3 82(2 3 5)5 5 105

25 3 55

Q37R2

532

72(3 3 7)2 5 32 3 72

QabR

m5

am

bm (b 2 0).(ab)m 5 ambm


Practising

3. Write each expression as a power with a single base. Showyour work.a) b) c) d)

4. Simplify. Express as a single power where possible.

a) d)

b) e)

c) f )

5. Evaluate.a) c)

b) d)

6. Multiple choice. Simplify A. B. C. D.

7. Multiple choice. Simplify A. B. C. D.

8. Multiple choice. Simplify .

A. B. C. D.

9. Kalyna can only enter one-digit numbers on her calculator. Theexponent key and the display are working fine. Explain how she canevaluate each power using her calculator.a) b)

10. Simplify to make it easier to evaluate. Show your work.

11. The side length of a cube is units.a) Determine the surface area of the cube without using powers.b) Determine the surface area using powers.c) Did you prefer the method you used in part a) or part b)?

Explain why.d) Determine the volume without using powers.e) Determine the volume using powers.f ) Did you prefer the method you used in part d) or part e)?

Explain why.

35

43 3 2503,

162254

55524512516

Q56

52R4

1.8101.861.8121.87(1.83 3 1.82)2.

29418224218(22 3 42)3.

(26 3 43)2

(23 3 42)2(32 3 12)2(32 3 12)3

Q55

53R3

(23 3 32)2

(25 3 52)2

(24 3 5)23(24) (33) 42(22 3 33)3

Q24

72R3

(43 3 32)2(45 3 32)3

Q46

44R3

(83 3 52)4

(24 4 3)3(9 4 3)2(4 3 6)3(3 3 7)2




8 cm

volume = 512 cm3

12. Navtej wants to paint her room and is on a budget. She found a 4 Lcan of paint, in a colour that she liked, on the mistints shelf at thehardware store. She knows that 500 mL covers 6 m2. She wants touse two coats of paint. Represent the area that she is able to paintusing a power. Recall that

13. Hye-Won is making ornamental paper lanterns for her ChineseNew Year party. Her first lantern is a cube.

1 L 5 1000 mL.

a) Express the volume of the lantern as a power.b) Another lantern has a volume of . How many times as

high is that cube than the first lantern?

14. Describe two different ways to evaluate Which wouldyou use? Why?

15. Suppose you are asked to evaluate and Whichexpression might you simplify first? Which one might you notsimplify? Explain.

Closing

16. Explain how can you simplify to calculate it using mentalmath.

Extending

17. a) Can you express as an equivalent power with a singlebase of 0.9, ? Explain how you know.

b) Can you express as an equivalent power with a base of0.81, ? Explain how you know.

c) When can you express a power with a base of 0.9 as anequivalent power with the base of 0.81?

18. Express each amount as a power with a single base. Show your work.

a) b) c) Q0.163

0.43 R3

(1.23 3 1.44)2(0.254 3 0.52)3

(0.9)3 5 (0.81)■

(0.9)3(0.81)3 5 0.9■

(0.81)3

403 3 55

105 3 83.28 3 254

63

23 .

215 cm3

Reading Strategy

Evaluating

Find someone whoused a different wayfrom you in questions14 and 15. Justify yourchoices to each other.


Communicate aboutCalculations with Powers

2.6


Clearly explain the steps for calculating with powers.

GOAL

INVESTIGATE the MathBay and Austin were answering this skill-testing question. Bay’s answerwas 1152 and Austin’s answer was 192. Austin started to show Bay whyhis answer was correct, but then his cell phone rang and he wasdistracted. Here is his explanation.

Use order of operations.

brackets firstthen exponentsdivide/multiply

Why is this question a good test of mathematical skill?

A. Use the Communication Checklist to help you improve andcomplete Austin’s explanation.

B. Why is this question a good test of mathematical skill?

Reflecting

C. Why is it important for an explanation to be complete and clear?

WORK WITH the Math

?

5 4 1

5 4 1 125 2 9 1 8 3 95 4 1 53 2 32 1 8 3 (3)2

4 1 53 2 32 1 8 3 (27 4 9)2

WIN A TRIP FOR 2 TO BANFF

Entry Form

Name:

e-mail:

Phone no:

Answer the following skill-testing question:

4�53�32+8�(27�9)2

YOU WILL NEED

• a calculator

✔ Did you include all thesteps?

✔ Did you explain why youdid each step?

✔ Did you explain how youdid each step?

✔ Did you justify yourconclusion?

Communication Checklist

Does ? Explain.

Austin’s Solution

62 1 65 5 67

I thought about what means and what means.67

62 1 65means

means6 3 6 3 6 3 6 3 6 3 6 3 6.67(6 3 6) 1 (6 3 6 3 6 3 6 3 6).62 1 65

EXAMPLE 1 Communicating about powers and exponents


NEL 772.6 Communicate about Calculations with Powers

I calculated to make sure. Sincepowers represent repeatedmultiplication, I did that beforeadding.My answer makes sense,because cannot bethe same as 36 3 7776.

36 1 7776

I think and are notequal, because is 36 times greaterthan not 36 more.

Check

so 62 1 652 67.7812 2 279 936,

67 5 279 936 5 7812 5 36 1 7776

62 1 65 5 36 1 65

65,67

6762 1 65

Calculate

Nicole’s Solution

52 1 316 3 (22 2 6) 4.

I used order of operations.I underlined the operations as I did them.

52 1 316 3 (22 2 6) 4

I need to evaluate the expression insidethe innermost brackets first. It containsan exponent and so does the first term. Ievaluated these powers. I then evaluatedthe expression in the round brackets bysubtracting. This left an expression insidethe square brackets which I evaluated bymultiplying.

5 25 1 316 3 (22) 45 25 1 316 3 (4 2 6) 4

52 1 316 3 (22 2 6) 4

I added the remaining numbers.

5 275 25 1 (232)

EXAMPLE 2 Simplifying using order of operations

You can use the memory aidBEDMAS to remember therules for order of operations.Perform the operations inBrackets first.Calculate Exponents andsquare roots next.Divide and Multiply from leftto right.Add and Subtract from left toright.

Communication Tip



I used order of operations to evaluate each expression in thenumerator and denominator.

5 1

5 1010

5 5 1 510

5 25 4 5 1 57 1 3

(42 1 32) 4 5 1 5

(42 2 32) 1 35

(16 1 9) 4 5 1 5(16 2 9) 1 3

EXAMPLE 3 Simplifying fractions using order of operations

Calculate

Derek’s Solution

(42

1 32) 4 5 1 5

(42 2 32) 1 3 .

Numerator: I evaluated the expression in the brackets byevaluating the powers then adding. I divided the result by fiveand then added five to this.Denominator: I evaluated the expression in the brackets byevaluating the powers then subtracting. I then added three to this.

I divided the numerator by the denominator.

Checking

1. Show the steps to evaluate each expression.a) b) c)

Practising

2. a) Evaluate b) Evaluate c) Would you use the expression in part a) or part b) for a

skill-testing question? Explain why.

32 1 4 3 22 2 10.32 3 4 1 22 2 10.

12 1 (26)2 4 392 1 9 4 324(3)2

In Summary

Key Idea

• When everyone follows the same order of operations, everyone gets thesame answer to a question.

Need to Know

• Use BEDMAS (Brackets, Exponents, Division, Multiplication, Addition,Subtraction) to remember the order of operations.• Evaluate the contents in brackets first, starting with the innermost

brackets.• Evaluate powers.• Multiply and divide from left to right.• Add and subtract from left to right.


NEL 792.6 Communicate about Calculations with Powers

3. Evaluate. Explain your strategy.a) b) c) d)

4. Evaluate.

a) c)

b)

5. a) Evaluate with a calculator.b) Does your calculator follow the order of operations? How do

you know?

6. Give an example of a product of two powers that is the same astheir sum. Explain how you came up with your example.

7. Explain why 2 is the only base for which

8. Is it possible for a power with a base of 5 to be equal to a powerwith a base of 10? Explain.

9. Which is greater: or ? How can you answer this without acalculator?

10. Larry is preparing meat and cheese skewers for a party. He has18 small skewers and 12 large skewers. Each small skewer needs 2 cubes of cheese. Each large skewer needs twice as many cubes of cheese.a) Which expression best describes how many cubes of cheese

Larry needs? Explain why.A. B.

b) Each small skewer needs 2 cubes of meat. Each large skewerneeds double the number of cubes of meat. Write an expression,using powers, to describe how many cubes of meat Larry needs.Explain your answer.

c) How many cubes of meat does Larry need?

Closing

11. Why is it important to use the order of operations when you usemathematics to communicate?

Extending

12. Ruby copied the solution to a math problem from the board duringclass. When she got home to review the problem, she spilled herjuice on her homework and couldn’t make out one of the exponentsin the question and part of the solution.a) Explain how Ruby can determine the missing exponent.b) Rewrite the original question and show all the steps to solve it.

13. To evaluate an expression involving a power, do you have to calculatethe power before multiplying and dividing? Explain using examples.

(18 3 2) 1 (12 3 22)18 3 22 1 12 3 2

320230

a2 2 a1 5 a2 4 a1.

122 1 52 2 64 4 42

52 2 5 4 5 1 2 2 1

(52 2 3) 3 2 4 11 1 332 2 (22 3 5)

43 1 32 3 4 4 2

73 2 2784 2 82122 1 4232 1 35

72 + 33 – 12 ÷ 22 = 732



2.7 Calculating Square Roots

EXAMPLE 1 Applying the Pythagorean theorem

Amanda’s Solution

Calculate the square roots of fractions and decimals.

GOAL

LEARN ABOUT the MathAmanda walks 0.5 km east to go to school. On Thursdays, she goes toYvonne’s house after school to play video games. Yvonne’s house is 1.2 kmsouth of the school. Amanda cuts through the park to get home.

How far does Amanda walk to get home from Yvonne’s house??

YOU WILL NEED

• a calculator

?park

Amanda’shouse

Yvonne’shouse

0.5 km

school

1.2 km

N

S

EW

I checked my answer by squaring.The square root is an exact valuesince squaring it results in thenumber I started with.

My answer seems reasonablebecause the distance across thepark is greater than 1.2 km.

I used the Pythagorean theoremto write a relationship betweenthe sides in the right triangle. Isolved for .c2c2 5 (0.5)2 1 (1.2)2

c2 5 a2 1 b2

The triangle is a right triangle.The distance across the park isthe hypotenuse, so it must begreater than 1.2 km.

I calculated the square root onmy calculator.

=2^31 . 1.69

c 5 1.3 km

�� =961 . 1.3

c 5 "1.69

c2 5 1.69 c2 5 0.25 1 1.44

It’s 1.3 km between our houses.

Different calculators calculatesquare roots in differentways. With some, you press

first and then enter thenumber. With others, youenter the number first andthen press . There areother ways too.

Technology Tip

a

bc

Amanda’s house school

Yvonne’shouse

park


NEL 812.7 Calculating Square Roots 81

Yvonne’s Solution: Using Fractions

I used the Pythagorean theorem towrite a relationship between thesides in the right triangle. I wrotethe decimals as fractions, where

and 1.2 .

Then I solved for .c 2

51210

0.5 55

10

c2 5 a2 1 b2

1310 3

1310 5

169100

c 5 1 310 or 1.3 km

c 51310

c 5!169!100

c 5 Å169100

c2 5169100

c2 525

100 1144100

c2 5 Q 510R

21 Q12

10R2

Then I evaluated the squareroots of the numerator and thedenominator.

My answer seems reasonable because this distance is greaterthan 1.2 km

It’s 1.3 km between our houses.

I checked my answer bymultiplying. My square root is anexact value.

Reflecting

A. How are Amanda’s and Yvonne’s methods similar? How are they different?

B. Do you prefer Amanda’s method or Yvonne’s method? Explain why.

C. Explain how Yvonne determined the square root of the fraction inher solution.

a

bc

Amanda’s house school

Yvonne’shouse

park

The triangle is a right triangle.The distance across the park isthe hypotenuse, so it must bethe longest side of the triangle.

I took the square root of both sides.The square root must be a

fraction, , where .

Since this is equivalent to ,

I reasoned that .!169!100

5ab

169100

5a2

b2

169100

5a 3 ab 3 b

ab


I calculated the side length ofa square with an area greaterthan 1. The square root is lessthan the number.

!1.21 5 1.1

1.21 1.1

I think this is going to happenfor all squares whose sideshave length greater than 0but less than 1.

The square root of a number isgreater than the number when thenumber is between 0 and 1.

I calculated the side length ofa square with an area of lessthan 1 square unit. Thesquare root is greater thanthe number.

!0.64 5 0.8

0.64 0.8


EXAMPLE 3 Determining the square root of a fraction using a quotient

Austin is building a patio using square concretepatio slabs; 25 of the slabs cover 9 . What are the dimensions of the top of each slab?

m2

WORK WITH the Math

EXAMPLE 2 Determining the square root of decimals greaterthan and less than 1

When is the square root of a number greater than the number?

Bay’s Solution

I know that the side lengthof a square with an area of1 square unit is 1 unit. I’ll trysquares with greater andlesser areas.!1 5 1

1 1


NEL 832.7 Calculating Square Roots

Since the top is a square, I know thelength and width must be equal. I cancalculate the length of each side bydetermining the square root of its area.

The top of each slab has an area of

The length of each side

is

5 0.6

535

Å9

25 5!9!25

Å9

25.

925.

I divided to determine thearea of one top.

The square root of a quotient is the sameas the quotient of the square roots.

I calculated the square root of thenumerator and the denominator.

The top of the slab hasdimensions of 0.6 m by 0.6 m.

I wrote the fraction as a decimal.

EXAMPLE 4 Using order of operations with a square root

Calculate

Amanda’s Solution

24 3 !36 1 42 4 2 1 1.

I treated the square root like a power.I evaluated the powers first.Then, I divided and multiplied.Lastly, I added.5 105

5 96 1 8 1 15 16 3 6 1 16 4 2 1 1

24 3 "36 1 42 4 2 1 1

In Summary

Key Idea

• If a positive number is less than 1, then its square root will be greaterthan the original number. If a positive number is greater than 1, then itssquare root will be less than the original number.

Need to Know

• The square root of a quotient equals the quotient of the square roots.

• If the numerator and denominator of a fraction are both perfectsquares, then the square root of the fraction is an exact value.

• If a decimal can be written as an equivalent fraction whose numeratorand denominator are perfect squares, then the square root of thedecimal is an exact value.

Åab 5

!a!b

Austin’s Solution



Checking

1. Enter the missing numbers.

a) c) e)

b) d) f )

2. A square field has an area of 1.44 . Calculate its length andwidth without a calculator. Show your work.

Practising

3. Enter the missing numbers.

a) b) c) d)

4. Calculate.

a) b) c)

5. Based on your answers to question 4, how can you predict the

answer to ?

6. Evaluate.

a) b)

7. Multiple choice. Evaluate

A. B. C. D.

8. Determine each square root to one decimal place.

a) c) e) g)b) d) f ) h)

9. Label the square and square root from each part of question 10 on anumber line. The first one is done for you.

10. The square root of 1 is 1. That is, Is any otherpositive rational number equal to its square root? How do you know?

11. The square root of a number is 16.5. What is the number?

12. You know that What decimal square roots could youcalculate easily using that information? Explain.

!576 5 24.

!1 5 !1 3 1 5 1.

1.0 1.20.80.60.40.20 1.4 1.6 1.8 2.0

2.56�

!0.25!0.4900!1.44!1.96!0.36!0.8100!1.69!2.56

1611

14 64165 536

121256

1116

Å121256.

(!81 1 !64)2 4 17 1 672 1 "4 3 42 2 2

Å6416

Å72981Å

819Å

91

!■!■

546Å

100289 5 ■!■ 5 0.07!3.61 5 ■

km2

!■!■

51013Å

■81 5

79!■ 5 11

!144!225 5

■■Å

49 5

■3"49 5 "■ 3 ■


NEL 852.7 Calculating Square Roots

radius

13. Bittu has a new TV with an 84 cm screen. He wants to put it above his fireplace in a space 150 cm wide and 75 cm high. Will the TV fit into the space?

14. a) Complete each statement.

A. B. C.

b) Explain how you know your answers are reasonable.

15. Verify each statement and correct those that are incorrect.a) c)b) d)

16. Calculate.a) b) c) d)

17. What pattern do you notice in the answers in the previous question?

18. What is the area of the yellow region?

Closing

19. Which of these have a square root that is an exact value: 0.49, 4.9,0.0049? Explain.

Extending

20. Could be a fraction with a denominator of 2? Explain.

21. The area of a circle with radius is , where What isthe radius of a circle of area 50.24 ?

22. The area of the square is 12.25 . Estimate the radius of the circle.

cm2

cm2p 8 3.14.pr 2r

Å■9

!0.000 000 09!0.000 009!0.0009!0.09

!0.25 5 0.5!0.9 5 0.03!256 5 16!6.4 5 3.2

!0.■ 51■

!0.04 51■Å

14 5 !0.■ 5

12

16 cm2

9 cm2



2.8 Estimating Square Roots

Use perfect square benchmarks to estimate square roots of otherfractions and decimals.

GOAL

INVESTIGATE the MathBay is preparing for the Egg Drop Experiment in science class. Bay willtry to drop the egg 23.7 m, without breaking it. He needs to determinehow long an egg will take to hit the ground. He will estimate the droptime for the egg using the formula , where time ismeasured in seconds and height in metres.

How long will it take an egg to hit the ground?

A. Substitute the known value into the formula.

B. What is the greatest perfect square less than the height? What is theleast perfect square greater than the height?

C. Which of the two numbers you found in part B is the given heightcloser to?

D. Estimate the square root of the height to one decimal place usingthe numbers from part B as benchmarks. Check your answer bymultiplying and estimate again if you need to.

E. Determine m to two decimal places using a calculator.

F. Write 23.7 as an improper fraction. Is the square root of 23.7 an exact value? Explain how you know.

G. How long will this egg take to hit the ground, to one decimal place?

Reflecting

H. Why is it helpful to estimate the square root of a number that is nota perfect square?

!23.7

?

time 5 0.45!height

YOU WILL NEED

• a calculator


NEL 872.8 Estimating Square Roots

WORK WITH the Math

Estimate

Nicole’s Solution

!0.84.

I thought of the decimal in hundredthsand looked for a square root that Iknew that was close to it.

0.84 is That is close to , and its squareroot is , or 0.9.9

10

81100

84100

I estimated 0.92 as the squareroot.

EXAMPLE 2 Estimating a square root by reasoning

Since I chose anumber a little greater than 0.9.

0.84 . 0.81,

0.92 3 0.92 5 0.8464

is about 0.92!0.84

I checked my estimate by squaring it.

My estimate is not an exact value.

Shelby knew that square root problems involve two identical numbers,so she said Is her answer reasonable?

Shelby’s Solution

!110 5 55.

I decided to estimate. I know 110isn’t a perfect square, so I thoughtof square numbers that are lessthan 110 and greater than 110.

100 110 121!110 5 55

EXAMPLE 1 Estimating a square root to verify a calculation

!121 5 11!100 5 10 I compared the square roots ofthese numbers to my estimate.

?

Obviously, so thesquare root of 110 is not 55. Theanswer is not reasonable.

3025 2 110,552 5 3025!110 5 55

is between 10 and 11, so myestimate of is not reasonable.!110 5 55!110

I squared the answer on mycalculator.

Yvonne’s Solution


The area of a square is between and . What might theside length of the square be?

Derek’s Solution: Using a number line

710 units23

10 units2

I needed a value between and

I looked for a number whose squareroot would be easy to calculate.

710.3

10

and so is 36100

710 5

70100,3

10 530

100

310

410

510

610

710

EXAMPLE 4 Identifying a square root between two numbers


Is either of these square roots an exact value: Evaluate each.

Bay’s Solution

!0.49, !4.9?

I can write 0.49 as a fraction wherethe numerator and denominator areperfect squares, so is anexact value.

!0.49

5710 or 0.7

!49!100

5

Å49

1005!0.49

0.7 3 0.7 5 0.49

EXAMPLE 3 Reasoning about square roots of decimals

I checked by multiplying.

so the square root of 4.9 is a decimal between 2 and 3.4 , 4.9 , 9

or

!490!100

5

Ä490100

5!4.9

!49!10

5

Å4910

5!4.9

I chose perfect square benchmarksof 4 and 9 to estimate !4.9.

I cannot write 4.9 as a fraction where thenumerator and denominator are perfectsquares.In my first try, the numerator is a perfectsquare, but the denominator is not.In my second try, the denominator is aperfect square, but the numerator is not,so is not an exact value.!4.9

�� =94 . 2.2135943

2.2 3 2.2 5 4.84

!4.9 8 2.2

Because 4.9 is much closer to 4 thanto 9, I estimated a decimal valueclose to 2.

I checked by multiplying.Then I compared my estimate to thevalue determined using a calculator.My estimate was reasonable.


between and .710

310

Å36100 5

610 I took the square root.

710 5 7 4 10 5 0.7

310 5 3 4 10 5 0.3

The side length of the square

might be units.610

I wrote and as decimals.710

310

�� =50 . .7071067

0.3 , 0.5 , 0.7

The side length of the square

might be units.710

I chose a number between 0.3 and0.7 and determined the square rootwith my calculator. Then I used anearby estimate.

Austin’s Solution: Using reasoning

NEL 892.8 Estimating Square Roots

Checking

1. List the two closest whole numbers between which each square root lies.

a) b) c) d)

2. Estimate each square root in question 1 to two decimal places usingyour calculator.

3. How do you know your answers to question 2 are reasonable?

Practising

4. Calculate the side length of a square with an area of 6.4 cm2.

Å59!149.7!52.4!8.5

In Summary

Key Idea

• You can use perfect squares as benchmarks to estimate the square rootof numbers that are not perfect squares. For example, to estimate

think that is 256 and is 289, so must be closer to16 than 17, or about 16.1.

Need to Know

• You can check the square root of a number by multiplying the squareroot by itself, or squaring it.

• Decimals that cannot be written as equivalent fractions with numeratorsand denominators that are both perfect squares have square roots thatare not exact values.

!259172162!259,


5. A square has an area of Estimate the side length of thesquare. Explain how you estimated.

6. The areas of some squares are shown. Estimate the length of thesides of each square. Then, determine the lengths using a calculator.

a) c) e)

b) d) f )

7. Multiple choice. Between which two whole numbers doeslie?

A. 25 and 30 B. 10 and 20 C. 5 and 6 D. none of these

8. Multiple choice. Calculate the side length of a square with an areaof A. 1.6 cm B. 40.96 cm C. 2.5 cm D. 0.8 cm

9. Pearl is going to paint her bedroom wall pink. The wall is 2.5 m by2.5 m. She has bought a can of paint that will cover a) Estimate to determine if she has enough paint for two coats.

Show your work.b) What is the side length of the largest square she can paint with

two coats? Answer to the nearest metre.

10. A square-based shed has a floor area of Which estimate iscloser to the length of the front of the shed: 7.2 m or 7.7 m?Explain how you can answer this without using a calculator.

11. a) How do you know that ?b) Will the square root of a decimal always be greater than the

square root of the decimal that is 0.1 greater? Explain.

12. Explain how you know that cannot be 0.8 or 0.08.

13. A baseball diamond is a square with a side length of about 27 m.Joe throws the ball from second base to home plate. Estimate howfar Joe threw the ball.

Closing

14. It’s sometimes easier to calculate the square root of a decimalhundredth than a decimal tenth without a calculator, for example,1.44 than 14.4. Is the same true for estimating?

Extending

15. Hedy estimated as 50. Explain how you could give a closerestimate.

16. The area of the rectangle is Divide the rectangle intosquares to determine the approximate length of each side. Describewhy you chose the strategy you used.

156 cm2.

!2358

!6.4

!0.7 . 0.8

50.6 m2.

20 m2.

6.4 cm2.

!26.7

3625 units21

4 units275.6 units2

16144 units20.01 units21.44 units2

31.5 cm2.


1st base

2nd base

3rd base

home plate

27m

A = 156 cm2


CHAPTER 2 Chapter Self-Test

NEL 91Chapter Self-Test

1. Sketch and label a representation for each of the following.a) b) c) a cube with a volume of 729

2. Krista has a set of stacking cubes. The numerical value of the volume of the largest cube is 40 times greater than the numerical value of the area of one face of the smallest cube. The smallest cube has a volume of 125 .a) Determine the side length and surface area of the

smallest cube. Show your work.b) Determine the side length and volume of the largest cube.

3. Represent each item as a power and evaluate it.a) b)

4. In Darrell’s DVD collection, there are 10 action movies, 18 comedies,4 cartoons, and 32 mysteries.a) Write an expression using powers to represent the number of

DVDs in each category.b) Write an expression using the product of two different powers

for the total number of DVDs.

5. Simplify as a single power and evaluate.a) c) e)

b) d) f )

6. Simplifya) b) c)

7. Solve using order of operations. Show each step.

8. Enter the missing numbers. Round to two decimal places if necessary.

a) c) e)

b) d) f )

9. A square garden has an area of 76 .a) Between which two whole numbers is the side length? Explain

how you determined these numbers without a calculator.b) Determine the side length to two decimal places.

WHAT DO You Think Now?Revisit What Do You Think? on page 49. Have your answers and explanations changed?

m2

!0.36 5 ■!■!■

567!■ 5 2.3

Å205 5 ■Å

81144 5

■■

!56 5 ■

3 3 4 1 (6 1 2)2 4 2

(c3)3 4 (c2)2(a4)2(x5) (x8) 4 x7

(25 3 32)2

(24 3 3)2(143)5 4 1415(27) (22)3

2(24 3 62)3(17)5 4 (1)5(24)3(24)5

(27) (27) (27) (27) (27) (27)2 3 2 3 2 3 2 3 2

cm3

cm31.22183

Be

V2

g

8

j


CHAPTER 2 Chapter Review


FREQUENTLY ASKED QuestionsQ: How can you simplify a power involving products and

quotients?

A: In a product, the exponent applies to each factor. For example, .In a quotient the exponent applies to both the numerator and

denominator. .

For example,

Q: How can you evaluate an expression involving many operations?

A: Use BEDMAS (Brackets, Exponents, Division, Multiplication, Addition, Subtraction) to help you remember the order to performthe operations. For example,

Evaluate what is in the Brackets. Startwith the innermost brackets, if there ismore than one set.

Evaluate powers next, using the Exponents.

Divide and Multiply from left to right.

Add and Subtract from left to right.

Q: How can you calculate or estimate a square root?

A1: You can use the square root key on your calculator Forexample,

You can check your answer by multiplying the square root by itselfto see if you get the original number.

A2: You can use perfect squares as benchmarks to estimate the squareroot of numbers that are not perfect squares. For example, is between and and is much closer to It is likelyabout 5.2.

!25.!36,!25!27.4

�� =4.72 5.234500931

(!).

5 34

5 4 1 30

5 4 1 60 4 2

5 4 1 6(10) 4 2

5 4 1 638 1 24 4 2

5 4 1 6323 1 24 4 2

4 1 6323 1 (6 2 4) 4 4 2

Q35R2

532

52.

QabRm

5am

bm

(2 3 3)5 5 25 3 35(ab)m 5 ambm

• See Lesson 2.5, Examples 1,2, 3, 4, and 5.

• Try Chapter Reviewquestion 12.

Study Aid

Study Aid

• See Lesson 2.7, Examples 1,2, and 3, and Lesson 2.8,Examples 1, 2, 3, and 4.

• Try Chapter Reviewquestions 15, 16, 17, 18,and 19.

Study Aid

• See Lesson 2.6, Examples1, 2, and 3 and Lesson 2.7,Example 4.

• Try Chapter Reviewquestions 13 and 14.


NEL 93Chapter Review

PracticeLesson 2.1

1. Sketch a model to represent the following. Label each side length.a) a square field with an area of b)c) a cube with a side length of three units

2. a) Calculate the side length of a square with an area of 196 .b) Calculate the side length of a cube with a volume of 125 .

3. Nita is planting 49 carrot seeds to grow in her garden. She wants toplant them in a square plot. She needs to plant them 3 cm apart,and 3 cm apart from the edge of the plot.a) Sketch the square garden with the seeds.b) Determine the dimensions of the garden.c) Determine the area of the garden.

Lesson 2.2

4. Complete the table.

cm3mm2

102

225 m2

Power Base Exponent Repeated Multiplication Value

25624

2(6) (6) (6)

(23)4

5. Evaluate without using a calculator. Show your work.a) b) c)

6. Susan needs to wrap two gift boxes in the shape of cubes. She has asheet of wrapping paper 140 cm by 30 cm. One box is 7 cm by 7 cmby 7 cm. Each side of the other box has an area of 529 cm2. Does shehave enough wrapping paper to wrap both boxes? Show your work.

Lesson 2.4

7. Simplify.a) b) c)

8. Evaluate.a) b) c)

9. Simplify.a) b) c) (v4)6 4 (v3)5a9 4 a5 4 a3(x 5) (x 2) 42

(232) (237)

(232)3(233)

(45)2

46(62) (63)2

(197) (19) 4 (192)2(192)(122)3

122(55)5(52)

2(21)322.3362

a)

b)

c)

The BestC A R R O T

The Best

$2.2050 seeds


3.2 m

? m4.8 m

10. Use repeated multiplication to explain why each statement is true.

a) b)

11. Express with a base of 2.

Lesson 2.5

12. Express as a power with a single base. Show your work.

a) b)

Lesson 2.6

13. Simplify without using a calculator. Show all your work.a) b)

14. Which question would you ask to see if someone understands orderof operations? Explain why.A. B.

Lesson 2.7

15. Evaluate.

a) d)

b) e)

c) f )

16. Verify each statement. Show your work.a) b) c)

Lesson 2.8

17. A square arena has an area of 200 .a) Without using a calculator, state the two whole numbers

between which its side length is located.b) Which whole number from part a) is a better estimate, and why?c) Determine the length of its side to two decimal places.

18. A square garden has an area of Which is a better estimate forthe length of the garden: 6.3 m or 6.9 m? Explain how you cananswer this without using a calculator.

19. Katie and her brother Nick started a window washing business to earnmoney in the summer. In one job, they had to wash windows thatwere 4.8 m off the ground. There was a hedge of large bushes besidethe house so they needed to set the base of the ladder 3.2 m away fromthe house. About how long did the ladder need to be?

40 m2.

m2

!0.0036 5 0.06!4.8 5 2.4!4.9 5 0.7

!70.8964Å1636

!255!39.69

!121!144!289

94 1 32 3 4394 3 32 1 43

62 1 2 3 32 2 8162 2 82 4 22

Q76

73R4

(63 3 364)2

322

62 3 65 5 6785

83 5 82



Testing Skills

A Mathletics Council is holding a contest to promote math awareness.The winners of the contest will share the $100 prize equally. To win, acontestant needs to answer a skill-testing math question correctly.You have been asked to create the question. Your question must meetthe following criteria.

• The answer to the question must be 100.• There must be at least four numbers in the question.• The operations must include a power and a square root.• To answer the question correctly, order of operations (BEDMAS)

must be used. If you just solve the question from left to right, you willnot get 100 as your answer.

• It must be tricky, because you want only a few people to win.• You must provide the correct solution to your question. • You must also provide examples of some mistakes that people might

make.

What is the best skill-testing question you can create??

Answer the skill-testing question

correctly and win $100!

CHAPTER 2 Chapter Task

YOU WILL NEED

• a calculator

✔ Did you show all the stepsin your solution(s)?

✔ Did you verify your solutions?

Task Checklist

NEL 95Chapter Task


mf9sb ch02 p46-69 pp7 - mr....

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