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Squares, Square Roots,Cubes & Cube Roots 10
CHAPTER 2SQUARES,SQUARE ROOTS.CUBES AND CUBE ROOTS
A. SQUARES
- a number multiply by itself- a2 = a a- examples :
a). 22 = 2 2 = 4
b). ( - 4 )2 = ( -4 ) ( -4 ) = 16
c). ( )5
3 2 = (5
3) (
5
3) =
25
9
d). ( 0.3 )2 = 0.3 0.3 = 0.09
- the square of any number is greater than zero and is always positive.
B. SQUARE ROOTS
- the square roots of any number is the number when multiplied by itself,equals to the given number.(inverse operation of squaring that number)
- If x = a2, then aaaax === 2
- examples :
a). 3339 ==
b).3
2
33
22
9
4=
=
c). 6.06.06.036.0 ==
- some fractions are required to reduce to the lowest terms in order tofind the square roots.
- examples:
a).32
3322
94
188 =
==
- to find the square roots of a mixed number, change the mixednumber into an improper fraction.
- example :
a).5
6
55
66
25
36
25
111 =
==
- The square root of negative numbers do not exist
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SQUARES SQUARE ROOTS
12 =1 1 = 1
22 = 4 4 = 2
32 = 9 9 = 3
42 = 16 16 = 4
52 = 25 25 = 5
62 = 36 36 = 6
72 = 49 49 = 7
82 = 64 64 = 8
92 = 81 81 = 9
102 = 100 100 = 10
112
= 121 121 = 11
122 = 144 144 = 12
132 = 169 169 = 13
142 = 196 196 = 14
152 = 225 225 = 15
162 = 256 256 = 16
172 = 289 289 = 17
182 = 324 324 = 18
192 = 361 361 = 19
202 = 400 400 = 20
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C. CUBES
- a number multiply by itself twice- a3 = a x a x a- examples :
a). 33 = 3 x 3 x 3 = 27
b). (3
2)3 =
27
8
3
2
3
2
3
2=
c). ( 0.2 )3 = 0.2 x 0.2 x 0.2 = 0.008
d). ( - 5 )3 = ( - 5 ) x ( - 5 ) x ( - 5 ) = - 125
- The cube of a positive number is positive- The cube of a negative number is negative.
D. CUBE ROOTS
- a number when multiply by itself twice, equal to the given number.
- aaaaa == 33 3
- examples :
a). 22228 33 ==
b).5
2
555
222
125
833 =
=
c). 6.06.06.06.0216.0 33 ==
d). 4)4()4()4(6433
==
- The cube root of a positive number is positive, the cube root of anegative number is negative.
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Module PMR
Squares, Square Roots,Cubes & Cube Roots 13
CUBES CUBE ROOTS
13 = 1 3 1 = 1
23 = 8 3 8 = 2
33 = 27 3 27 = 3
43 = 64 3 64 = 4
53 = 125 3 125 = 5
63
= 216 3 216 = 6
73 = 343 3 343 = 7
83 = 512 3 512 = 8
93 = 729 3 729 = 9
103 = 1000 3 1000 = 10
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Squares, Square Roots,Cubes & Cube Roots 14
QUESTIONS :
A. Find the value of the following.
1). 32 = 2). 62 =
3). 82 = 4). 92 =
5). 112 = 6). 122 =
7). ( - 2 )2 = 8). ( - 4 )2 =
9). ( - 5 )2 = 10). ( - 7 )2 =
11). ( - 9 )2 = 12). ( - 10 )2 =
13).
2
2
1
= 14).
2
5
2
=
15).
2
7
3
= 16).
2
5
11
=
17).
2
9
4
= 18).
2
3
11
=
19).
2
3
23
= 20).
2
12
7
=
21). ( 0.4 )2 = 22). ( 1.2 )2 =
23). ( - 0.3 )2 = 24). ( - 0.05 )2 =
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Squares, Square Roots,Cubes & Cube Roots 15
B. Find the value of the following.
1). 4 = 2). 25 =
3). 64 = 4). 81 =
5). 100 = 6). 144 =
7). 225 = 8). 196 =
9).64
1= 10).
25
4=
11).100
9= 12).
9
71 =
13).16
91 = 14).
9
111 =
15).4
112 = 16).
162
50=
17).49462 = 18).
25214 =
19. 64.0 = 20. 0025.0 =
21. 21.1 = 22. 25.2 =
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C. Find the values of the following:
1). 23 = 2). 43 =
3). 73 = 4). ( - 5 )3 =
5). ( - 3 )3 = 6). 103 =
7).
3
52
= 8).
3
43
=
9).
3
6
1
= 10).
3
4
11
=
11).
3
3
21
= 12).
3
10
7
=
13). ( 0.1 )3 = 14). ( 0.6 )3 =
15). ( - 0.2 )3 = 16). ( - 0.03 )3 =
17). ( 1.2 )3 = 18). ( - 0.4 )3 =
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Squares, Square Roots,Cubes & Cube Roots 17
D. Find the value of the following.
1). 3 8 = 2). 3 27 =
3). 3 216 = 4). 3 125 =
5). 3 512 = 6). 3 343 =
7). 3 1000 =8). 3
8
1=
9). 364
27= 10). 3
8
33 =
11). 3125
1000= 12). 3
64
611 =
13). 3 343.0 = 14). 3 000216.0 =
15). 3 064.0 = 16). 3 125.0 =
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Squares, Square Roots,Cubes & Cube Roots 18
Common Errors.
Questions Errors Correct Steps
1. a). Find the value
0f 3 125 .
b).Calculate the value of
3 648
1 2 .
a). (-5) x (-5) x (-5)or
5 P 0
b).
2
48
1
=
2
2
1
K 0
=
2
1
2
1
=4
1N 0
a). 5 1m
b). ( )2
48
1
=2
2
1
1m
=
2
1
2
1
=4
11m
2. a). Find the value of3 216.0 .
b).Calculate the value of3
116
25
.
a). 0.006 P 0
b). 314
5
=1
1
4
5
=4
4
4
5 K 0
=4
1N 0
a). 0.6 1m
b).
3
14
5
=
3
4
4
4
5
=
3
4
1
1m
=64
11m
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3. a). Find the value of
3
3
1
.
b). Calculate the value of
( )16
92
3
a).
3
1
3
1
3
1
or
27
1P 0
b). 8 x16
9K 0
=2
9
= 2
14 N 0
a).
27
11m
b). ( )4
38
= ( ) 32 1m
= 6 1m
Questions based on PMR format
1. a). Find the value of
2
3
1
.
b). Calculate the value of ( )3
836 .
2. a). Find the value of 3 008.0 .
b). Calculate the value of 16 3 27 .
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Module PMR
Squares, Square Roots,Cubes & Cube Roots 20
3. a). Find the value of 3 216.0 .
b). Calculate the value of 38
27
2
1 .
4. a). Find the value of 81.0 .
b). Calculate the value of ( )23 275.4 .
5. a). Find the value of 3 343 .
b). Calculate the value of 15 3 64 .
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Squares, Square Roots,Cubes & Cube Roots 21
6. a). Find the value of
3
4
1
.
b). Calculate the value of16
1
64
9 .
7. a). Find the value of25
241 .
b). Calculate the value of 22 129 + .
8. a). Find the value of9
17 .
b). Calculate the value of ( )23 1443 .
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Squares, Square Roots,Cubes & Cube Roots 22
9. a). Find the value of (- 0.4)2 .
b). Calculate the value of ( )2255.5 .
10. a). Find the value of
3
5
1
.
b). Calculate the value of 52 x 3125
216 .
11. a). Find the value of4
120 .
b). Calculate the value of 102 3 1000 .
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Squares, Square Roots,Cubes & Cube Roots 23
12. a). Find the value of 3 216.0 .
b). Calculate the value of ( )23.081.0 + .
PMR Past Years Questions
2004
a). Find the value of 3 512.0 .
b). Calculate the value of 42 x 38
27 . ( 3 marks )
2005
a). Find the value of
3
4
1
.
b). Calculate the value of ( )2
3 272.4 . ( 3 marks )
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Squares, Square Roots,Cubes & Cube Roots 24
2006
a). Find the value of 49.0 .
b). Calculate the value of
3
1
16
25
( 3 marks )
2007
a). Find the value of 3 64 .
b). Calculate the value of
3
362
1
. ( 3 marks )
2008
a). Find the value of 3 27
1
.
b). Calculate the value of ( )28116 . ( 3 marks )
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Squares, Square Roots,Cubes & Cube Roots 25
CHAPTER 2 : SQUARES ROOTS,CUBES,&CUBE ROOTSANSWERS
A.
1). 9 2). 36
3). 64 4). 81
5). 121 6). 144
7). 4 8). 16
9). 25 10). 49
11). 81 12). 100
13).4
114).
25
4
15).49
916).
25
36=
25
111
17).81
1618).
9
16=
9
71
19). 9
4
139
121
= 20). 144
49
21). 0.16 22). 1.44
23). 0.09 24). 0.0025
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Squares, Square Roots,Cubes & Cube Roots 26
B.
1). 2 2). 5
3). 8 4). 9
5). 10 6). 12
7). 15 8). 14
9). 8
1
10). 5
2
11).10
312).
3
11
3
4=
13).4
11
4
5= 14).
3
13
3
10=
15).2
13
2
7= 16).
9
5
17).7
51
7
12= 18).
5
12
5
11=
19). 0.8 20). 0.05
21). 1.1 22). 1.5
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Squares, Square Roots,Cubes & Cube Roots 27
C.
1). 8 2). 64
3). 343 4). 125
5). 27 6). 1000
7).125
88).
64
27
9).216
110).
64
611
64
125=
11).27
174
27
125= 12).
1000
343
13). 0.001 14). 0.216
15). 0.008 16). 0.00027
17). 1.728 18). 0.064
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Squares, Square Roots,Cubes & Cube Roots 28
D.
1). 2 2). 3
3). 6 4). 5
5). 8 6). 7
7). 108).
2
1
9).4
310).
2
11
2
3=
11).5
10= 2 12).
2
11
2
3=
13). 0.7 14). 0.5
15). 0.4 16). 0.5
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Squares, Square Roots,Cubes & Cube Roots 29
No. Marking Scheme Marks
1.a).
9
1
b). ( - 2 )3
- 8
1
1
1= 3
2.a). 0.2
b). 16 + 3
19
1
1
1= 3
3.a). 0.6
b).2
3
2
1+
22
4=
1
1
1= 3
4.
a). 0.9
b). ( 1.5 )2
2.25
1
1
1= 3
5.a). 7
b). 15 + 4
19
1
1
1= 3
6.a).
64
1
b). 8
3
1
4
2
11
2
3=
1
1
1= 3
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Squares, Square Roots,Cubes & Cube Roots 30
7.
a).5
21
5
7=
b). 225
15
1
1
1= 3
8.a).
3
22
3
8=
b). 152
225
1
1
1= 3
9. a). 0.16
b). ( 1.1)2
1.21
1
1
1= 3
10.a).
125
1
b).
5
6
25
- 30
1
1
1= 3
11.a).
2
14
2
9=
b). 100 + 10
110
1
1
1= 3
12.a). 0.6
b). ( 1.2 )2
1.44
1
1
1= 3
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Squares, Square Roots,Cubes & Cube Roots 31
2004a). 0.8
b).2
316
- 24
1
1
1= 3
2005a).
64
1
b). ( 1.4)2
1.96
1
1
1= 3
2006a). 0.7
b).
3
4
1
64
1
1
1
1= 3
2007a). 4
b). ( 3 )3
27
1
1
1= 3
2008a).
3
1
b). 72
49
1
1
1= 3