Transcript
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
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