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Master 4.9
Lesson 4.2 Check Your Understanding
Calculators are not permitted in chapter 4
MIAP ch 4.2 Classifying Numbers outcome A2a pgs 207-210
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Master 4.9
Lesson 4.2 Check Your Understanding
Let’s try a few more numbers…
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Master 4.9
Lesson 4.2 Check Your Understanding
yes
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Master 4.9
Lesson 4.2 Check Your Understanding
4.2 Irrational numbers pg 207-211 outcome A2a
1. Tell whether each number is rational or irrational. Explain how you know.
a)
49
16
b)
3
30
c) 1.21
Solution
a)
b)
3
49
16
is rational since
49
16
=
7
,
4
49
16
is a perfect square.
or 1.75
30 is irrational since –30 is not a perfect cube.
The decimal form of
3
30 neither repeats nor terminates.
c) 1.21 is rational since it is a terminating decimal.
Homework Pg 211
#3-4
#5,10,12
#21
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Master 4.9
Lesson 4.2 Check Your Understanding
2. Use a number line to order these numbers from least to greatest.
2, 3 2, 3 6, 11, 4 30
Solution
2 is between the perfect squares 1 and 4, and is closer to 1.
2
4
1
↓
↓
↓
1
?
2
2  1.4142
–2 is between the perfect cubes –1 and –8, and is closer to –1.
3
3
3
1
2
8
↓
↓
↓
–1
?
–2
3
2  1.2599
6 is between the perfect cubes 1 and 8, and is closer to 8.
3
3
3
1
6
8
↓
↓
↓
1
?
2
3
6  1.8171
11 is between the perfect squares 9 and 16, and is closer to 9.
11
16
9
↓
↓
↓
3
?
4
11  3.3166
30 is between the perfect fourth powers 16 and 81, and is closer to 16.
4
4
4
16
30
81
↓
↓
↓
2
?
3
4
30  2.3403
Mark each number on a number line.
From least to greatest:
3
2,
2,
3
6,
4
30,
11
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Master 4.10
Lesson 4.3 Check Your Understanding
4.3 Mixed and Entire Radicals Outcome A2b
Pg 213-217
1. Simplify each radical.
a) 63
b) 3 108
c)
4
128
Solution
Write each radical as a product of prime factors, then simplify the radical.
a)
63  3  3  7
 3 3  7
3 7
b) 3 108  3 2  2  3  3  3
 3 2  2  3 3 3 3
 3 22 3
 33 4
c) 4 128  4 2  2  2  2  2  2  2
 4 2222  4 222
 24 8
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Master 4.10
Lesson 4.3 Check Your Understanding
2. Write each radical in simplest form, if possible.
a) 30
b) 3 32
c) 4 48
Solution
Look for perfect nth factors, where n is the index of the radical.
a) The factors of 30 are: 1, 2, 3, 5, 6, 10, 15, 30
There are no perfect square factors other than 1.
So, 30 cannot be simplified.
b) The factors of 32 are: 1, 2, 4, 8, 16, 32
The greatest perfect cube is 8.
8 = 2 · 2 · 2, so write 32 as 8 · 4.
3
32  3 8  4
 3 83 4
 23 4
c) The factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
The greatest perfect fourth power is 16.
16 = 2 · 2 · 2 · 2, so write 48 as 16 · 3.
4
48  4 16  3
 4 16  4 3
 24 3
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Master 4.10
Lesson 4.3 Check Your Understanding
3. Write each mixed radical as an entire radical.
a) 7 3
b) 2 3 4
c) 2 5 3
Solution
a) Write 7 as:
7  7  49
7 3  49  3
 49  3
 147
b) Write 2 as:
3
2 2 2  3 8
5
2  2  2  2  2  5 32
23 4  3 8  3 4
 3 8 4
 3 32
c) Write 2 as:
2 5 3  5 32  5 3
 5 32  3
 5 96
Homework pg 218
#3-6
#10-12
#24
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Master 4.11
Lesson 4.4 Check Your Understanding
4.4 Fractional Exponents page 222-226 outcome A3
1. Evaluate each power without using a calculator.
a) 1000
1
3
b) 0.25
1
2
c) ( 8)
1
3
d)
 
16
1
4
81
Solution
The denominator of the exponent is the index of the radical.
1
a) 1000 3  3 1000
 10
1
b) 0.25 2  0.25
 0.5
1
c) ( 8) 3  3 8
 2
d)
 
16
81
1
4
 4 16
81
2

3
2
2. a) Write 26 5 in radical form in 2 ways.

65 and
b) Write
4
19

3
in exponent form.
Solution
 a
m
a) Use a n 
2
26 5 
n

5
26
m
or

2
or
n
5
am .
262
m
b) Use
am  a n .
n
5
65  6 2
Use
 
n
m
a
 
4
3
19
m
 an .
3
 19 4
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Master 4.11
Lesson 4.4 Check Your Understanding
3. Evaluate.
4
3
b) ( 27) 3
a) 0.012
3
c) 814
d) 0.751.2
Solution
3
1


a) 0.012   0.012 




0.01

3
3
 0.13
 0.001
1


b) (27)  (27) 3 


4
3


3
27
  3
 81
 1
c) 81   814 


3
4


4
81


4
4
4
3
3
 33
 27
d) 0.751.2
Use a calculator.
0.751.2  0.7080
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Master 4.11
Lesson 4.4 Check Your Understanding
2
4. Use the formula b  0.01m 3 to estimate the brain mass of each animal.
a) a moose with a body mass of 512 kg
b) a cat with a body mass of 5 kg
Solution
2
Use the formula b  0.01m 3 .
a) Substitute: m = 512
2
b  0.01(512) 3
b  0.01( 3 512)2
Use the order of operations.
Evaluate the power first.
b  0.01(8)2
b  0.01(64)
b  0.64
The brain mass of a moose is approximately 0.64 kg.
b) Substitute: m = 5
2
b  0.01(5) 3
Use a calculator.
b  0.0292
The brain mass of a cat is approximately 0.03 kg.
Homework pg 227
#3-7
#12,16,19
4.5
Negative Exponents pages 229-232 outcome A3
Homework pa233
#3,4,7,8
#9,10,13
#21
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Master 4.13
Lesson 4.6 Check Your Understanding
4.6 Exponent laws pages 237-241 outcome A3
1. Simplify by writing as a single power. Explain your reasoning.

b) 

a) 0.82  0.87
1.5 
3 5
c)
 

4
5
5
4
d)
5
1.5
9 9
9
2
3


  
 

4
5
4


5
1
4
3
4
Solution
a) 0.82  0.87
Use the product of powers law: When the bases are the same, add the exponents.
0.82  0.87  0.82  7
 0.85

 
1
0.85
3
 
3
 
5
 4 2
 4 4
  
b)  

 5 
 5 
First use the power of a power law: For each power, multiply the exponents.


 

4


  
2
5

4
4
5
5
 
 
     

 


4
(2)( 3)
5
4

6

5

4
4
(4)( 5)
5
20
5
Then use the quotient of powers law.


 

4
3


  
2
5
 

4
5
4


5
 
  



1.5 
4
6  20
5
4
14
5
3 5
c)
5
Use the power of a power law.
1.5

1.5
( 3)( 5)
5
1.5

15
1.5
5
Use the quotient of powers law.
1.5
 1.515  5
 1.510
(Solution continues.)
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Master 4.13
5
4
9 9
d)
9

9
Lesson 4.6 Check Your Understanding
1
4
Use the product of powers law.
3
4
5 1
4 4
9
3
4
4

9
9
4
Use the quotient of powers law.
3
4
4
 94

3
4
1
 94
2. Simplify. Explain your reasoning.
a) m4 n2  m2 n3
b)
6 x4 y 3
14 xy 2
Solution
a) m4 n2  m2 n3  m4  m2  n2  n3
 m4  2  n2  3
Use the product of powers law.
 m6 n
b)
6 x4 y 3
14 xy 2

6 x4

14 x

y 3
y2
Use the quotient of powers law.
3
7
  x 4  1  y 3  2
3
7
 x3 y 5

3 x3
7 y5
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Master 4.13
Lesson 4.6 Check Your Understanding
3. Simplify. Explain your reasoning.
3
1
 


b)  x3 y 2  x 1 y 2 



3
a) (25a 4 b 2 ) 2
c)
1
5
2
12 x 5 y
 50 x2 y 4  2
d)  4 7 
 2x y 
1
1
2
3x y
2
Solution
3
3
a) (25a 4b2 ) 2  25 2  a
 3
4 
 2
b
3
2 
2
Using the power of a power law.
3
 1
  25 2   a 6  b3


 53  a6b3
 125a6b3
3
1
3
1
 



b)  x3 y 2  x 1 y 2   x3  x 1  y 2  y 2



 x3  1  y

Use the product of powers law.
3 1

2 2
 x2  y 1
x2
y

c)
12 x 5 y
1
2
3x y
5
2
1
2
12 x 5
  1
3
2
x
 4 x
 4 x

5 

11
2

y
Use the quotient of powers law.
1
y
1
2
5
2
2
5
 y2

1
2
 y3
4 y3
x
11
2
(Solution continues.)
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Master 4.13
Lesson 4.6 Check Your Understanding
d) Simplify inside the brackets first.
1
1
 50 x2 y 4  2  50 x2 y 4  2
 2 x4 y7    2  x4  y7 

 

  25  x 2  4  y 4  7  2
1
  25  x 2  y 3  2
1
1
 25 2  x
1
( 2) 
2
 5  x 1  y


y
Use the power of a power law.
1
( 3) 
2
3
2
5
xy
3
2
4. A cone with height and radius equal has volume 18 cm3. What are the radius and height
of the cone to the nearest tenth of a centimetre?
Solution
1
3
The volume V of a cone with radius r and height h is given by the formula: V  r 2 h
Because the height and radius are equal, substitute: h = r
The formula becomes:
1
3
1 3
r
3
V  r 2 (r )
V
Substitute V = 18, then solve for r.
1
3
18  r 3
3(18)  3
Multiply each side by 3
 r 
1
3
3
54  r 3
Divide each side by .
r

3
54


54

 r3
To solve for r, take the cube root of each side by raising
each side to the one-third power.
1
 54  3
 
 
  r 3 3
1
1
 54  3
 
 
r
r  2.5807
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Master 4.13
Lesson 4.6 Check Your Understanding
The radius and height of the cone are approximately 2.6 cm.
Homework 241
#3-6
#10,15,16
#21,22
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