Chapter 9
Exercises page 106
12, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
2307, 5207, 223, 421, 719, 613 are prime numbers.
345 = 9 x 5 = 32 x 5
54 = 9 x 6 = 32 x 3 x 2 = 33 x 2
63 = 9 x 7 = 32 x 7
92 = 2 x 2 x 23 = 22 x 23
95 = 5 x 19
121 = 11 x 11 = 112
364 = 2 x 2 x 7 x 13 = 22 x 7 x 13
428 = 2 x 2 x 107 = 22 x 107
1526 = 2 x 7 x 109
1533 = 3 x 7 x 73
4a- a = 105 = 5 x 3 x 7
b = 70 = 2 x 5 x 7
GCD(a, b) = 5 x 7 = 35
LCM(a, b) = 2 x 3 x 5 x 7 = 210
b- a = 1321 = 1321
b = 16 = 24
GCD(a, b) = 1
LCM(a, b) = 1321 x 24 = 21136
c- a = 2206 = 2 x 1103
b = 1418 = 2 x 709
GCD(a, b) = 2
LCM(a, b) = 2 x 709 x 1103 = 1564054
d- a = 3064 = 23 x 383
b = 520 = 23 x 5 x 13
GCD(a, b) = 23 = 8
LCM(a, b) = 23 x 5 x 13 x 383 = 199160
5a- 17/10 is decimal
5/2 = 25/10 is decimal
19/5 = 38/10 is decimal
21/5 = 42/10 is decimal
b- 7/20 = 35/100 is decimal
5/8 = 625/1000 is decimal
c- 127/625 = 2032/10000 is decimal
63/8 = 0.375 = 375/1000
9/32 = 0.28125 = 28125/100000
27/25 = 1.08 = 108/100
42/70 = 0.6 = 6/10
35/112 = 0.3125 = 3125/10000
63/125 = 0.504 = 504/1000
8a- 19/5 = 3
8/10
b- 53/12 = 4
416/1000
648/72 = 9
0
375/306 = 1
225/1000
690/222 = 3
108/1000
1098/63 = 17
428/1000
914/9 = 1.55
17/3 = 5.66
78/99 = 0.78
543/999 = 0.543
30/99 = 0.30
17/9 = 1.88
11a- √12 = √22 x 3 = 2√3
√18 = √32 x 2 = 3√2
√45 = √9 x 5 = 3√5
√48 = √24 x 3 = 4√3
b- √27 = √33 = 3√3
√75 = √52 x 3 = 5√3
√125 = √53 = 5√5
√72 = √23 x 32 = 6√2
c- √300 = √3 x 102 = 10√3
√1000 = √103 = 10√10
√4000 = √22 x 103 = 20√10
√16000 = √42 x 103 = 40√10
d- √8 x √32 = √8 x 32 = √23 x 25 = √28 = 24 = 16
√45 x √5 = √45 x 5 = √32 x 5 x 5 = √32 x 52 = 3 x 5 = 15
√12 x √48 = √12 x 48 = √22 x 3 x 24 x 3 = 23 x 3 = 24
e- √7 x √5/7 = √7 x 5/7 = √5
√9 x √25/36 = √32 x 52/32 x 22 = 5/2
√(15/11 x 33/20) = √((3 x 5 x 3 x 11)/(11 x 22 x 5)) = 3/2
12a- √11 x √22 x √50 = √(11 x 22 x 50)
= √(11 x 2 x 11 x 2 x 52)
= 11 x 2 x 5 = 110
√75 x √15 x √20 = √(75 x 15 x 20)
= √(52 x 3 x 5 x 3 x 22 x 5)
= 52 x 3 x 2 = 150
b- √17 x √34 x √32 = √(17x34x32)
= √(17x17x2x25)
= 17 x 23 = 136
√28 x √72/11 x 45/77 = √(28x72/11x45/77) = 144/11√10
c- √5/7 x √5/7 x √5/7 = √(5/7x5/7x5/7) = 5/7√5/7
√40 x √16/33 x √12/11 = √(40x16/33x12/11)
= √((23x5x24x22x3)/(3x11x11))
= 24/11√10
d- √4x104 = 2 x 102
√9x106 = 3 x 103
√25x10-2 = 5 x 10-1
13225 = 52 x 32 = (5x3)2 = 152
625 = 54 = (52)2 = 252
perfect square
perfect square
441 = 32 x 72 = (3x7)2 = 212
perfect square
450 = 52 x 2 x 32
not a perfect square
5625 = 54 x 32 = (52x3)2 = 752 perfect square
676 = 22 x 132 = (2x13)2 = 262 perfect square
3025 = 52 x 112 = (5x11)2 = 552
perfect square
5400 = 23 x 52 x 33
not a perfect square
14a- √75 = √(52x3) = 5√3
√405 = √(5x34) = 9√5
√112 = √(24x7) = 4√7
√225 = √(52x32) = 15
b- √1225 = √(52x72) = 35
√1296 = √(24x34) = 22 x 32 = 36
√1875 = √(54x3) = 25√3
√25200 = √(24x52x32x7) = 60√7
c- √576 = √(26x32) = 23 x 3 = 24
√1089 = √(32x112) = 3 x 11 = 33
√3679 = √(13x283)
√4032 = √(26x32x7) = 24√7
d- √2925 = √(52x32x13) = 15√13
√2025 = √(52x34) = 5 x 32 = 45
√6075 = √(52x35) = 5x32√3 = 45√3
√1352 = √(23x132) = 26√2
15x
576
2116
3893.76
1369
15129
59049
24448.4496
√x
24
46
62.4
37
123
243
156.36
x
529
552.25
3283.29
7921
20164
6185.8225
126906.9376
16a- √1127 = 33.57
b- √14076 = 118.64
c- √6348 = 79.67
d- √4567.56 = 67.58
17a- a = √5
b = 2√5
b=2xa → b>a
b- a = -√7
b =√2
a < 0 and b > 0
→b>a
c- a = √13
b = √17
17 > 13
→ √17 > √13
→b>a
√x
23
23.5
57.3
89
142
78.65
356.24
d- a = 2√5
b = 3√2
a2 = (2√5)2 = 20
b2 = (3√2)2 = 18
a2 > b2 → a > b
e- a = 2 + √7
b = 2√7
a – b = 2 + √7 - 2√7 = 2 - √7 < 0
→a<b
f- a = 3 - √7
b = 3 - √5
a – b = 3 - √7 – 3 + √5
= √5 - √7 < 0
→a<b
g- a = 3 + 2√3
b = 3 + √2 + √5
a – b = 3 + 2√3 – 3 - √2 - √5
= 2√3 - √2 - √5 < 0
→a<b
h- a = 11 + √2
b = 5 + 2√2
a – b = 11 + √2 – 5 - 2√2
= 6 - √2 > 0
→a b
i- a = √7 + 2
b = 2√7 + 1
a – b = √7 + 2 - 2√7 – 1
= -√7 + 3 > 0
→a>b
j- a = 2√3 + 3√5
b = √3 + 4√5
a – b = 2√3 + 3√5 - √3 - 4√5 = √3 - √5 < 0
→a<b
18a- x = 4√3 - 5√12 + √300
= 4√3 - 5√22x3 + √3x102
= 4√3 - 10√3 + 10√3
= 4√3
y = 2√5 + √45 - √80
= 2√5 + √32x5 - √24x5
= 2√5 + 3√5 - 4√5
= √5
x2 = (4√3)2 = 16x3 = 48
y2 = (√5)2 = 5
x2 > y2
x>y
b- x = √11 - 5√44 + 4√99
= √11 - 5√22x11 + 4√32x11
= √11 - 10√11 + 12√11
= 3√11
y = √5 - 4√45 + 7√20
= √5 - 4√32x5 + 7√22x5
= √5 - 12√5 + 14√5
= 3√5
3√11 > 3√5
→x>y
c- x = -2√7 - 3√28 + 6√63
= -2√7 - 3√22x7 + 6√32x7
= -2√7 - 6√7 + 18√7
= 10√7
y = 2√50 + √72 - √128
= 2√52x2 + √23x32 – v27
= 10√2 + 6√2 - 8√2
= 8√2
x2 = (10√7)2 = 700
y2 = (8√2)2 = 128
x2 > y2
x>y
d- x = 2√12 - 3√75 + 4√27
= 2√22x3 - 3√52x3 + 4√33
= 4√3 - 15√3 + 12√3
= √3
y = √44 + √99
= √22x11 + √32x11
= 2√11 + 3√11
= 5√11
5√11 > √3
y>x
19a- (1 + √5)(1 - √5) = 12 – (√5)2
= 1 – 5 = -4
b- (1 + √6)( √2 - √3) = √2 - √3 + √12 - √18
= √2 - √3 + 2√3 - 3√2
= -2√2 + √3
c- (2 - √6)( √2 - √3) = 2√2 - 2√3 - √12 + √18
= 2√2 - 2√3 - 2√3 + 3√2
= 5√2 - 4√3
d- (√2 + √5)( √2 - √5) = (√2)2 – (√5)2
= 2 – 5 = -3
20a- 1 - √3
1 + √3
b- √7
√7
c- √5 + √7
√5 - √7
d- 5 + √7
√7 – 5
e- 2 - √11
2 + √11
f- √13 – 4
√13 + 4
g- 2√3 + 3√2
2√3 - 3√2
h- 4 + √12
4 - √12
i- √10 - √2
√2 + √10
j- 5√5 + 6√6
6√6 - 5√5
21a- 1/√7 x √7/√7 = √7/7
b- 1/(1+√3) x (1-√3)/(1-√3) = 1-√3/(1-3) = -1+√3/2
c- 5/(2+√11) x (2-√11)/(2-√11) = (10-5√11)/-7 = (-10+5√11)/7
d- 2/(1-√5) x (1+√5)/(1+√5) = (2+2√5)/(1-5) = (-1-√5)/2
e- √2/(5-√2) x (5+√2)/(5+√2) = (5√2 +2)/3
f- √3/(2+√3) x (2-√3)/(2-√3) = (2√3-3)/(2-3) = -2√3 + 3
g- 2√5/(2+√5) x (2-√5)/(2-√5) = (4√5–10)/(2-5) = (-4√5 + 10)/3
h- (1-√2)/(1+√2) x (1-√2)/(1-√2) = (1-2√2+2)/-1 = -3 + 2√2
i- (5+√7)/(5-√7) x (5+√7)/(5+√7) = (25+10√7+7)/-2 = -16 - 5√7
j- 3√7/(2+√7) x (2-√7)/(2-√7) = (6√7-21)/(2-7) = (-6√7+21)/5
k- 1/(√5+√7) x (√5-√7)/( √5-√7) = (√5-√7)/(5-7) = (-√5+√7)/2
l- √2/(2+3√5) x (2-3√5)/(2-3√5) = (2√2-3√10)/-13 = (-2√2+3√10)/13
m- (√2+√3)/(2√3+3√2) x (2√3-3√2)/(2√3-3√2) = (2√6-12+6-3√6)/(6-18) = (√6+6)/12
n- (1-√3)/(1+√4) x (1-√4)/(1-√4) = (1-√4-√3+√12)/-3 = (1-√3)/3
o- (2+√5)/(2√3+√12) = (2+√5)/(2√3+2√3) = (2+√5)/(4√3) x √3/√3 = (2√3+√15)/12
p- 1/(√12+√27) = 1/(2√3+3√3) = 1/5√3 x √3/√3 = √3/15
q- 1/(√20+√45) = 1/(2√5+3√5) = 1/5√5 x √5/√5 = √5/25
r- 1/(√28+√7) = 1/(2√7+√7) = 1/3√7 x √7/√7 = √7/21
Chapter 9
Problems page 109
1a- (1 + √5)(2 - √5) = 2 - √5 + 2√5 – 5
= -3 + √5
b- (1 + √6)( √2 + √3) = √2 + √3 + √12 + √18
= √2 + √3 + 2√3 + 3√2
= 4√2 + 3√3
c- (2 - √6)( √2 + √3) = 2√2 + 2√3 - √12 -√18
= 2√2 + 2√3 - 2√3 - 3√2
= -√2
d- (√2 + √3)2 + (√10 – 1) = (√2)2 + 2(√2)(√3) + (√3)2 + √10 – 1
= 2 + 2√6 + 3 + √10 – 1
= 4 + 2√6 + √10
e- (3 - √5)2 + (√5 + √2)2 = 32 – 2(3)(√5) + (√5)2 + (√5)2 + 2(√5)(√2) + (√2)2
= 9 - 6√5 + 5 + 5 + 2√10 + 2
= 21 - 6√5 + 2√10
f- (√3 - √7)( √3 + √7) = (√3)2 – (√7)2
= 3 – 7 = -4
2a- 3√72 + 2√64 - √18 + 1 = 3√36x2 + 2√16x4 - √9x2 + 1
= 18√2 + 16 - 3√2 + 1
= 17 + 15√2
b- 4√75 + 2√27 - √36 + 3 = 4√25x3 + 2√9x3 - √6x6 + 3
= 20√3 + 6√3 – 6 + 3
= 26√3 – 3
c- 3√20 + 7√45 + √25 + 2 = 3√4x5 + 7√9x5 + √5x5 + 2
= 6√5 + 21√5 + 5 + 2
= 27√5 + 7
d- √700 + 3√28 - 3√81 = √7x100 + 3√7x4 - 3√9x9
= 10√7 + 6√7 – 27
= 16√7 – 27
3a- √75 - 3√48 + 3√12 = √25x3 - 3√16x3 + 3√4x3
= 5√3 - 12√3 + 6√3
= - √3
b- 2√20 - 4√45 + 3√180 = 2√4x5 - 4√9x5 + 3√36x5
= 4√5 - 12√5 + 18√5
= 10√5
c- √700 - 4√28 + 9√7 = √100x7 - 4√4x7 + 9√7
= 10√7 - 8√7 + 9√7
= 11√7
d- 5√12 - 4√108 + 7√27 = 5√4x3 - 4√36x3 + 7√9x3
= 10√3 - 24√3 + 21√3
= 7√3
e- √500 + 2√45 - 7√20 = √5x100 + 2√9x5 - 7√4x5
= 10√5 + 6√5 - 14√5
= 2√5
4-
x
√x
X2
9/49
3/7
81/2401
6a- x2 = 9
x2 – 9 = 0
(x – 3)(x + 3) = 0
x–3=0
or
x=3
x+3=0
x = -3
b- x2 = 25
x2 – 25 = 0
(x – 5)(x + 5) = 0
x–5=0
or
x=5
x+5=0
x = -5
c- x2 = 1
x2 – 1 = 0
(x – 1)(x + 1) = 0
x–1=0
or
x=1
x+1=0
x = -1
d- x2 – 36 = 0
(x – 6)(x + 6) = 0
x–6=0
or
x=6
x+6=0
x = -6
25/4
5/2
625/16
5/9
√5/3
25/81
e- x2 – 49 = 0
(x – 7)(x + 7) = 0
x–7=0
or
x=7
x+7=0
x = -7
f- x2 – 15 = 0
(x – √15)(x + √15) = 0
x – √15= 0
or
x + √15= 0
x = √15
x = -√15
g- x2 + 16 = 0
→ x2 = -16
impossible
h- x2 + 1 = 0
→ x2 = -1
impossible
i- x2 – 7 = 0
(x – √7)(x + √7) = 0
x – √7= 0
or
x = √7
x + √7= 0
x = -√7
a- x2 = 2500
x2 – 2500 = 0
(x – 50)(x + 50) = 0
x – 50 = 0
or
x = 50
x + 50 = 0
x = -50
b- x2 – 169 = 0
(x – 13)(x + 13) = 0
x – 13 = 0
or
x = 13
x + 13 = 0
x = -13
7-
c- -3x2 = -108
→ x2 = -108/-3 = 36
x2 – 36 = 0
(x – 6)(x + 6) = 0
x–6=0
or
x=6
x+6=0
x = -6
d- (x + 1)2 = 4
(x + 1)2 – 4 = 0
[(x + 1) – 2][(x + 1) + 2] = 0
(x – 1)(x + 3) = 0
x – 1 = 0 or x + 3 = 0
x = 1 or
x = -3
e- x2 + 2x + 1 = 0
(x + 1)2 = 0
(x + 1) = 0
x = -1
f- x2 + 4x = -4
x2 + 2x + 4 = 0
(x + 2)2 = 0
(x + 2) = 0
x = -2
g- x2 + 4x + 4 = 16
(x + 2)2 = 16
(x + 2)2 – 16 = 0
[(x + 2) – 4][(x + 2) + 4] = 0
(x – 2)(x + 6) = 0
x – 2 = 0 or x + 6 = 0
x = 2 or
x = -6
h- x2 – 4x = 12
x2 – 4x – 12 = 0
x2 – 4x + 4 – 4 – 12 = 0
(x – 2)2 – 16 = 0
[(x - 2) – 4][(x - 2) + 4] = 0
(x – 6)(x + 2) = 0
x – 6 = 0 or x + 2 = 0
x=6
or
x = -2
8a- (√5 – 2) solution ??
x2 + 4x – 1 = (√5 – 2)2 + 4(√5 – 2) – 1
= 5 - 4√5 + 4 + 4√5 – 8 – 1
=0
→ (√5 – 2) is a solution
b- (1 + √7)2 = 1 + 2√7 + 7 = 8 + 2√7
c- (1 + √7)2 + a(1 + √7) – 6 = 0
8 + 2√7 + a(1 + √7) – 6 = 0
a(1 + √7) = -2 – 2V7
a = (-2 -2√7)/(1 + √7) x (1 - √7)/(1 - √7)
= -12/6
= -2
9a- area = πr2
547.1136 = πr2
r2 = 547.1136/π = 174.24
r = + √174.24 = 13.2cm
b- 0.5476 = πr2
r2 = 0.5476/π = 0.174
r = √0.174 = 0.417cm
c- 7.065 = πr2
r2 = 7.065/π = 2.25
r = √2.25 = 1.5cm
(radius is always positive)
10a- (x + 1)2 – (x – 1)2 = (x2 + 2x + 1) – (x2 – 2x + 1)
= x2 + 2x + 1 – x2 + 2x – 1
= 4x
b- 212 – 192 = (20 + 1)2 – (20 – 1)2
= 4(20)
= 80
c- 262 – 242 = (25 + 1)2 – (25 – 1)2
= 4(25)
= 100
2
151 – 1492 = (150 + 1)2 – (150 – 1)2
= 4(150)
= 600
2
2001 – 19992 = (2000 + 1)2 – (2000 – 1)2
= 4(2000)
= 8000
200012 – 199992 = (20000 + 1)2 – (20000 – 1)2
= 4(20000)
= 80000
11a- (x + a)2 – (x – a)2 = (x2 + 2xa + a2) – (x2 – 2xa + a2)
= x2 + 2xa + a2 – x2 + 2xa – a2
= 4xa
b-
222 – 182 = (20 + 2)2 – (20 – 2)2
= 4(2)(20)
= 160
342 – 262 = (30 + 4)2 – (30 – 4)2
= 4(4)(30)
= 480
752 – 652 = (70 + 5)2 – (70 – 5)2
= 4(5)(70)
= 1400
1282 – 1242 = (126 + 2)2 – (126 – 2)2
= 4(2)(126)
= 1008
12(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2

300012 = (30000 + 1)2 = (30000)2 + 2(30000)(1) + 12
= 900000000 + 60000 + 1
= 900060001

19992 = (2000 + 1)2 = (2000)2 + 2(2000)(1) + 12
= 4000000 + 4000 + 1
= 3996001

20092 = (2000 + 9)2 = (2000)2 + 2(2000)(9) + 92
= 4000000 + 36000 + 81
= 3964081

19752 = (2000 + 25)2 = (2000)2 + 2(2000)(25) + 252
= 4000000 + 100000+ 625
= 3900625

40252 = (4000 + 25)2 = (4000)2 + 2(4000)(25) + 252
= 16000000+ 200000 + 25
= 16200625
13area(square) = area(triangle)
s2 = (bxh)/2
s2 = (12x9)/2 = 54
s = √54 = 3√6 cm = 3√6 x 10 mm
14l = 3w
l x w = 210
→ 3w x w = 210
3w2 = 210
→ w2 = 210/3 = 70
→ w = √70
15a- AB + BC = 2√3 + √75
= 2√3 + 5√3
= 7√3
= √49x3 = √147 = AC
16
INK right triangle → IK2 = IN2 + NK2 = 12 + 22 = 5
→ IK = √5

KC = IC – IK = 7 – V5

(KA) // (CE) → IK/IC = IA/IE
→ IE = (IC x IA)/IK = 7x4/√5 = 28/√5 x √5/√5 = 28√5/5

AE = IE – IA = 28√5/5 – 4 = (28√5 – 20)/5

(KA) // (CE) → IK/IC = KA/CE → CE = (7x4)/ √5 = 28/√5 = 28√5/5