REVIEW - Ms. Turnbull's Website - Home

R E V I E W

1. Evaluate each radical. Why do you not need a

calculator?

a) 3J1000

c)6764

b) JOM

2. Explain, using examples, the meaning of the

index of a radical.

3. Estimate the value of each radical to 1 decimal

place. What strategies can you use?

a) JTl b) 3 f : U c) 4 / l 5

4. Identify the number in each case.

a) 5 is a square root of the number.

b) 6 is the cube root of the number.

c) 7 is a four th root of the number.

5. For 735, does its decimal fo rm terminate,

repeat, or neither? Support your answer wi th

an explanation.

m

6. Tell whether each number is rational or

irrational. Justify your answers.

a) - 2 b) 17 c) J\6

d) /32~ e) 0.756 f) 12.3

g ) 0 h) V81 i) TT

7. Determine the approximate side length of a

square wi th area 23 cm 2 . How could you check

your answer?

8. Look at this calculator screen.

9.

a) Is the number 3.141 592 654 rational or

irrational? Explain.

b) Is the number TT rational or irrational?

Explain your answer.

Place each number on a number line, then

order the numbers f r o m least to greatest.

^30, /20, Vl8, 3 f z 3 0 , 730, 4 / l 0

10. The formula T = 2ir^ ^ gives the time,

T seconds, for one complete swing o f a

pendulum wi th length L metres. A clock

pendulum is 0.25 m long. What time does

the pendulum take to complete one swing?

Give the answer to the nearest second.

11. Write each radical in simplest form,

a) /T50 b) y i 3 5

c) J u l d) 4/l62~

12. Write each mixed radical as an entire radical,

a) 6/5 b) 3 /II :)473 d) 21/2

13. Alfalfa cubes are fed to horses to provide

protein, minerals, and vitamins.

Two sizes of cubes have volumes 32 c m 3 and

11 cm 3 . What is the difference in the edge

lengths of the cubes? How can you use radicals

to f ind out?

246 Chapter 4: Roots and Powers

14. A student simplified /300 as shown:

7300 = 73 • /Too = 73 • 750 • 750 = 73 • 72 • 725 • Jl • Jl5

= 3 - 5 - / 2 - 5

= 75/2 Identify the errors the student made, then write

a correct solution.

15. Arrange these numbers in order f rom greatest

to least, without using a calculator. Describe

your strategy.

5/2,4/3, 3/6, 277, 6/2

16. Show, wi th examples, why a" — "Ja, when n is a

natural number and A is a rational number.

17. Express each power as a radical. I 5

,4 a) IT b) (-50) 3

,0.5

a) i f f c) 1.2

18. Express each radical as a power,

a) JlA b) \f\32

c) ( ^ 5 ) '

19. Evaluate each power without using a calculator.

'4/2 \ 3

a) 16 0.25 b) 1.445

d) (—X2

,16,

20. Radioactive isotopes decay. The half-life o f an

isotope is the time for its mass to decay by -j.

For example, polonium-210 has a half-life of

20 weeks. So, a sample of 100 g would decay to

50 g in 20 weeks. The percent, P, o f polonium

remaining after time t weeks is given by the

formula P = 100(0.5)2°. What percent of

polonium remains after 30 weeks?

21. Arrange these numbers in order f rom greatest

to least. Describe the strategy you used.

4 75,5l 3 /5,5l( v /5) 3

22. Kleiber's law relates a mammal's metabolic rate

while resting, q Calories per day, to its body

mass, M kilograms: 3

q = 70M 4

What is the approximate metabolic rate of

each animal?

a) a cow wi th mass 475 kg

b) a mouse wi th mass 25 g

23. a) Identify the patterns in this list.

81 = 3 4

27 = 3 3

9 = 3 2

b) Extend the patterns in part a downward.

Write the next 5 rows in the pattern.

c) Explain how this pattern shows that a~n = —

when a is a non-zero rational number

and n is a natural number.

24. Evaluate each power without using a calculator.

^2V3 . / 4 s

a) 2 -2 b) c) 25

25. Kyle wants to have $1000 in 3 years. He uses this

formula to calculate how much he should invest

today in a savings account that pays 3.25%

compounded annually: P = 1000(1.0325)~3

How much should Kyle invest today?

26. A company designs a container wi th the shape

of a triangular prism to hold 500 mL of juice.

The bases of the prism are equilateral triangles

wi th side length s centimetres. The height,

h centimetres, of the prism is given by the

formula: 1

h = 2000(3) 2 S " 2

What is the height of a container wi th base side

length 8.0 cm? Give your answer to the nearest

tenth of a centimetre.

Review 247

27. W h e n musicians play together, they usually tune

their instruments so that the note A above

middle C has frequency 440 Hz, called the

concert pitch. A formula for calculating the

frequency, F hertz, of a note n semitones above

the concert pitch is:

F= 440(12

v/2)"

Middle C is 9 semitones below the concert

pi tch. What is the frequency of middle C? Give

your answer to the nearest hertz.

28. Simplify. Explain your reasoning,

a) (3m 4 n) 2

c) (16 a2b6) 5 d) r3s~l

2^-2

29. Simplify. Show your work.

a) (a3b)(a~lb4) b) (x*y}(xY

c) — • a a5

d) x2y

x Y 2

30. Evaluate.

b)

c) 1 2 | 3

5 d)

(~5-5)3

( - 5 . 5 ^

3

0.16 4

0.164

31. A sphere has volume 1100 cm 3 . Explain how to

use exponents or radicals to estimate the radius

of the sphere.

32. Identify any errors in each solution, then write

a correct solution.

1 ~

:-ly-3//VM>| _ c - l . c 4 . ,3 . +3 a) \ s - ' f 3 / (5 4 f 3 )

At b) —

= s'4t

-I2c~l

= -I2c~l

1

~ 12c

THE WORLD OF MATH

Historical Moment: The Golden Ratio

The rat io, 1 +J5

: 1, is called the golden ratio.

Buildings and pictures w i t h dimensions in this ratio

are o f ten considered visually pleasing and "na tura l . "

The Greek sculptor Phidias used the golden rat io for the

dimensions of his sculptures. His 42-ft . high statue of the

Greek god Zeus in the temple in Olympia, created in

about 435 B . C . E . , was one of the Seven Wonders of the

Ancient Wor ld . The number 1 +/5

2 is of ten called "ph i 1

after the first Greek letter in "Phidias."

248 Chapter 4: Roots and Powers

C h a p t e r 4 : Review, page 246

1. a) 10

c) 2

b) 0.9

2. The index tells which root to take.

3. a) 3.3 b) -2.3

c) 2.0

4. a) 25 b) 216

c) 2401

5. Neither

6. a) Rational

c) Rational

e) Rational

g) Rational

i) Irrational

7. Approximately 4.8 cm

8. a) Rational

9. IPSO , VlO , Vl8 , ^30 , >/20 , >/30

(-30 ip) ^30

H I I I \ H

b) Rational

d) Irrational

f) Rational

h) Irrational

b) Irrational

10. 1 s

11. a) 5V6

c) 4^7

12. a) Vl80

c) vT92

13. Approximately 1.0 cm

15. 6>/2 , 3^6, 5V2 , 473, 2V7

17. a) VT2

c) Vh2

18. a) 1.42

4 c) 2.5s

19. a) 2

b) 3 /̂5

d) 3</2

b) y/\26

d) ^32

b) ^ / ( -50) 5 , or (v̂ ioy

V8

c) -32

20. Approximately 35%

21. , 5 \ 5~\ ^ 5 , </5

b) 133

<> ( t f b) 1.2

27 d) —

64

22. a) Approximately 7122 Calories/day

b) Approximately 4 Calories/day

23. a) The numbers at the left are divided by 3 each time; the

exponents in the powers at the right decrease by 1 each

time.

b) 3 = 3'; 1=3° ; - =3"'; - = 3"2; — =3~ 3

3 9 27

24. a) -4

. 125 c)

' 8

25. $908.51

26. 18.0 cm

27. 262 Hz 28. a) 9 m V

b) 27

c) Aab"

29. a) a V

c> 7 9

30. a) -4

v 144 c)

' 25

b) - h x y

b) -y

1

d) * V

b) 30.25

d) 0.4 31. Approximately 6.4 cm

32. a) sV

b) 64c

Chapter 4: Pract ice T e s t , page 249

1. B

2. A

3. a) 5^3 ; 5 V J = V75

b)

fiO 5^3

| I I I 'I I I ' l I I I 8.0 9.0

4. a) i ' 3

c) 0.729

' 16

A N S W E R S 491

492 A N S W E R S