Download Additional Example 10.1

New Progress in Junior Mathematics 2B
Rational and Irrational Numbers
10
Solution:
Rational and Irrational Numbers
(a) 12 
Additional Example 10.1
Find the square roots of each of the
following numbers.
(b) 7
(a) 100
(b) 2.56
(c)
(c) 

378 189

100 50
Additional Example 10.3
x
Express 0.9 5 in the form
, where x and
y
Solution:
(a)  10 10  100 and
(10)  (10)  100
 Square roots of 100   10

6 7  7  6 55


7
7
7
(c) 3.78 
144
25
(b) 
12
1
y are integers and y  0.
Solution:
Let x  0.9 5  0.959595... .......................(1)
1.6 1.6  2.56 and
(1.6)  (1.6)  2.56
Square roots of 2.56   1.6
then 100 x  95.9595... ............................(2)
(2) – (1):
100 x  x  95.9595...  0.959595...
99 x  95
12 12 144
 
and
5 5
25
 12   12  144
   
 5   5  25
144
12

Square roots of
25
5
x

95
0.9 5 
99
Additional Example 10.4
Determine whether the following numbers
are rational or irrational numbers. Explain
your answers briefly.
Additional Example 10.2
Express the following numbers in the form
x
, where x and y are integers and y  0.
y
(a)
(a) 12
158
(b) 441
(c) π  π
6
(b) 7
7
(c) 3.78
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New Progress in Junior Mathematics 2B
Rational and Irrational Numbers
Solution:
(a) 

(b) 

(c) 

Solution:
158  12.5698... which is neither
a terminating decimal nor a
recurring decimal.
158 is an irrational number.
80
80

36
36

441  21 21  21
441 is a rational number.
4 5
6
2 5

3
Additional Example 10.7
Simplify the following expressions.
Additional Example 10.5
(a) Express 243 and 675 as a product of
their prime factors in index notation.
243 and
62

π  π  2π  6.2831... which is
neither a terminating decimal nor a
recurring decimal.
π  π is an irrational number.
(b) If a  3 , express
in terms of a.
42  5
(a) 3 6  10 6
(b) 2 5  9 5
675
Solution:
(a) 3 6  10 6  13 6
Solution:
(a) 243  3  3  3  3  3
 35
675  3  3  3  5  5
(b) 2 5  9 5  7 5
Additional Example 10.8
Simplify the following expressions.
 33  52
(b)
(a)
(b)
243  35
 (32 ) 2  3
 32 a
Solution:
 9a
675  3  5
3
63  28
150  54
(a)
2
 32  3  52
63  28  32  7  22  7
3 7 2 7
5 7
 3 3 5
 15a
(b)
Additional Example 10.6
Express
2 6
80
in its simplest form.
36
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150  54  52  6  32  6
5 6 3 6
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New Progress in Junior Mathematics 2B
Rational and Irrational Numbers
Solution:
Additional Example 10.9
Simplify the following expressions.
(a)
25  75  ( 120 )
(b)
64  40  32
(a)
 3 6 (3 7  6 )  7 (3 7  6 )
 (3 6 )(3 7 )  (3 6 )( 6 )
 ( 7 )(3 7 )  ( 7 )( 6 )
 9 42  18  21  42
Solution:
 3  8 42
25  75  ( 120 )
(a)
(3 6  7 )(3 7  6 )
 52  3  52  ( 23  3  5 )
(b)
 5  5 3  (2 2  3  5 )
( 7  3 )2
 ( 7 ) 2  2( 7 )( 3 )  ( 3 ) 2
 5  5  (2)  3  2  3  5
 7  2 21  3
 50  2  3  5
2
 10  2 21
 50  3 10
  150 10
Additional Example 10.11
Express the following in their simplest form.
64  40  32
(b)
(a)
64  40

32

64  40
32
5
11
(b)
Solution:
(a)
 80
 42  5
4 5
5
5
11


11
11
11
5  11

11  11

Additional Example 10.10
Simplify the following expressions.
(b)
(a) (3 6  7 )(3 7  6 )
(b) ( 7  3 ) 2
5 11
11
3
3
 2
40
2 10
3

2 10

3
10

2 10
10
30
2 10
30

20

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New Progress in Junior Mathematics 2B
Rational and Irrational Numbers
Additional Example 10.12
Simplify
52 4 2
.

6
18
Solution:
52 4 2
52
4 2



2
6
6
18
23

52 4 2

6
3 2

52
2 4 2


6
3 2
2
52 2 4 2

6
6
56 2

6
28 2

3

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