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N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m M C 1 1 Q l d - 1 1
Exercise 1A — Classification of numbers 1 a 4 = 2, 4 is rational ⇒ Q
b 45
is rational ⇒ Q
c 79
is rational ⇒ Q
d 2 = 1.414 213 562…, 2 is non–terminating and non–recurring ⇒ I
e 7 = 2.645 751 311, 7 is non–terminating and non–recurring ⇒ I
f 0.04 = 0.2, 0.04 is rational ⇒ Q
g 2 12
is rational ⇒ Q
h 5 = 2.236 067 977… is non–terminating and non–recurring decimal ⇒ Ι
i 94
is rational ⇒ Q
j 0.15 is a terminating decimal ⇒ Q k −2.4 is a terminating decimal ⇒ Q l 100 = 10, 100 is rational ⇒ Q m 14.4 = 3.794 733 192… is a non–terminating and non–
recurring decimal ⇒ I n 1.44 = 1.2 1.44 is rational ⇒ Q o π = 3.141 592 654… is a non–terminating and non–
recurring decimal ⇒ I
p 25 59 3
= , 259
is rational ⇒ Q
q 7.32 is a terminating decimal ⇒ Q r − 21 = − 4.582 575 695…, − 21 is a non–terminating
and non–recurring decimal ⇒ I s 1000 = 31.622 776 6…, 1000 is a non–terminating
and non–recurring decimal ⇒ I t 7.216 349 157… is a non–terminating and non–recurring
decimal ⇒ I u − 81 = −9, − 81 is rational ⇒ Q v 3π = 9.424 777 961… is a non–terminating and non–
recurring decimal ⇒ I w 3 62 = 3.957 891 61…, 3 62 is a non–terminating and
non–recurring decimal ⇒ I
x 1 116 4
= is rational ⇒ Q
y 3 0.0001 0.046 415 88...= is a non–terminating and non–recurring decimal ⇒ I
2 a 18
is rational ⇒ Q
b 625 = 25, 625 is rational ⇒ Q
c 114
is rational ⇒ Q
d 08
= 0, 08
is rational ⇒ Q
e −6 17
is rational ⇒ Q
f 3 81 = 4.326 748 711…, 3 81 is a non–terminating and non–recurring decimal ⇒ I
g − 11 = − 3.316 624 79…, − 11 is a non–terminating and non–recurring decimal ⇒ I
h 1.444
= 1.22
= 0.6, 1.444
is rational ⇒ Q
i π = 1.772 453 851…, π is a non–terminating and non–recurring decimal ⇒ I
j 80
is undefined
k 3 21 = 2.758 924 176… 3 21 is a non–terminating and non–recurring decimal ⇒ I
l 7π = 0.448 798 95…,
7π is a non–terminating and non–
recurring decimal ⇒ I
m ( )2 33 5 25− = = 2.924 017 738…, ( )23 5− is a non–terminating and non–recurring decimal ⇒ I
n 311
− is rational ⇒ Q
o 1 1100 10
= , 1100
is rational ⇒ Q
p 6416
is rational ⇒ Q
q 2 2 0.282 842 71...25 5
= = is a non–terminating and
non–recurring decimal ⇒ I
r 6 1.224 744 87...2
= is a non–terminating and non–
recurring decimal ⇒ I s 3 27 is rational ⇒ Q
t 1 124
= is rational ⇒ Q
u 22 9.873 576 91...7π is a non–terminating and non–
recurring decimal ⇒ I v 3 1.728 1.2− = − is rational ⇒ Q w 6 4 6 2 12= × = is rational ⇒ Q
x ( )42 4= is rational ⇒ Q y 4 6 9.797 958 97...= is a non–terminating and non–
recurring decimal ⇒ I 3 A π = 3.141 592 654…, π is a non–terminating and non–
recurring decimal ⇒ I
B 4 29 3
= , 49
is a rational number ⇒ Q
C 912
= 0.866 025 403…, 912
is a non–terminating and
non–recurring decimal ⇒ I D 3 3 = 1.442 249 57…, 3 3 is a non–terminating and
non–recurring decimal ⇒ I The answer is B.
4 A − 81 = −9, − 81 is a rational number ⇒ Q
B 65
is a rational number ⇒ Q
C 3 343 = 7, 3 343 is a rational number ⇒ Q
Chapter 1 — Number systems: the Real Number System
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M C 1 1 Q l d - 1 2 N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m
D 0.0676 = 0.26, 0.0676 is a rational number ⇒ Q E 22 = 4.690 415 76…, 22 is a non–recurring and
non–terminating decimal ⇒ I The answer is E.
5 −0.69 is a rational number 7 = 2.645 751 311…, 7 is an irrational number
3π = 1.047 197 557…,
3π is an irrational number
49 = 7, 49 is a rational number The answer is C. 6 2½ is a rational number
113
− is a rational number
624 24.979 991 9...= is an irrational number 3 99 4.626 065 009...= is an irrational number The solution is D. 7 a 5 is a positive whole number, 5∈Z+
b 0.621 is a terminating decimal, 0.621∈Q
c 1 1 0.181 9
= = & ⇒ is a recurring decimal, 181
∈Q
d 0.26& is a recurring decimal 0.26& ∈Q e 3 + 16 = 3 + 4 = 7 ⇒ a positive whole number, 3
+ 16 ∈Z+ f 0.5151… = 0.51&& is a recurring decimal, 0.5151…∈Q g 3 8 = 2 ⇒ a positive whole number, 3 8 ∈ Z+
h 426
= 7 ⇒ a positive whole number, 426
∈ Z +
i 7 4 = 7 × 2 ⇒ a positive whole number, 7 4 ∈ Z +
j − 273
= −3 is a negative whole number, − 273
∈ Z −
k 9 − 144 = 9 − 12 = −3 is a negative whole number, 9 − 144 ∈Z −
l 04
= 0, 04
∈Q
m ( )24− − = −4 is a negative whole number, ( )24− − ∈Z −
n 93
= 3 is a positive whole number 93
∈Z +
o 23 9 2.25
2 4⎛ ⎞− = − = −⎜ ⎟⎝ ⎠
is a terminating decimal
232
⎛ ⎞−⎜ ⎟⎝ ⎠
∈Q
p 0.421& & is a recurring decimal, 0.421& &∈Q
q 8 4 22
= = is a positive whole number, 82
∈Z +
r 5 32− = −2 is a negative whole number, 5 32− ∈Z −
s ( )26− = 6 is a positive whole number, ( )26− ∈Z + t 6
5− = −1.2 is a terminating decimal, 6
5− ∈ Q
u 8 2 16 4− × = − = − is a negative whole number, 8 2− × ∈Z −
v 4 1.33
= & is a recurring decimal, 43
∈Q
w ( )62 = 8 is a positive whole number, ( )62 ∈Z + x 100 10 5
2 2= = is a positive whole number, 100
2∈Z +
y 3 343− = −7 is a negative whole number, 3 343− ∈Z − 8 a 6 is a positive whole number, 6 ∈Z+
b 0.3425… is a non–recurring and non–terminating decimal, 0.3425…∈ I
c 7 2.645 751 31...= is a non–recurring and non–terminating decimal, 7 ∈ I
d 9 3 0.7516 4
= = is a terminating decimal, 916
∈ Q
e 2 25 2 5 10− = − × = − is a negative whole number, 2 25− ∈ Z −
f 6 2 12 3.464 101 615× = = is a non–recurring and non–terminating decimal, 6 2× ∈ I
g 49 7− = − is a negative whole number, 49− ∈ Z − h 21 5 46.957 427 53...× = is a non–recurring and non–
terminating decimal, 21 5× ∈ I i 0.612612… = 0.612& & is a recurring decimal,
0.612612…∈ Q j 0.25 is a terminating decimal, 0.25∈ Q
k 144 16 49
= = is a positive whole number, 1449
∈Z +
l 3 64 4− = − is a negative whole number, 3 64− ∈ Z −
m 1119
is a rational number, 1119
∈ Q
n 9 1 1144 16 4
= = is a rational number, 9144
∈ Q
o 50 25 52
= = is a positive whole number, 502
∈Z +
p 5π = 15.707 963 27 is a non–recurring and non–terminating decimal, 5π ∈ I
q 16 × 3 27− = 16 × −3 = −48 is a negative whole number, 16 × 3 27− ∈ Z −
r ( )33 3 3 5.196 152 42...= = is a non–recurring and non–terminating decimal, ( )33 ∈ I
s 7 5 35 5.916 079 78...× = = is a non–recurring and non–terminating decimal, 7 5× ∈ I
t 6 3 16 6 3 4 72− × = − × × = − is a negative whole number, 6 3 16− × ∈ Z−
u 16 2 1.414 213 56...8
− = − = − is a non–recurring and
non–terminating decimal, 168
− ∈ I
v 8 12.5 100 10× = = is a positive whole number. 8 12.5× ∈Z+
w 15
− is a rational number, 15
− ∈ Q
x ( )2 9.869 604 40...π = is a non–recurring and non–terminating decimal, ( )2π ∈ I
y ( )3 125 5 5− − = − − = is a positive whole number, 3 125− − ∈Z +
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N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m M C 1 1 Q l d - 1 3 9 ( )13 8
12
7 2
7 2
7 18
+
= + ×
= +=
8 is a positive whole number. The answer is C
10 3
3
144 5129 816 64
4 416
×
= ×= ×=
terminating nor recurring ⇒ π ∈ Q′
16 is a positive whole number. The answer is C. 11 16 is a positive whole number ⇒ 16∈ Z+ (and rational) −3 2 = 4.242 640 7… is neither terminating nor recurring
⇒ −3 2 ∈I. 0 is a whole number but is neither positive nor negative ⇒
0 ∈ Z (and rational). π = 3.141 592… is neither positive nor negative ⇒ 0 ∈ Ι
8 4 22
= = = is a positive whole number ⇒ 82
∈Z+
(and rational) The answer is B. 12 2 + 9 = 10.414 213… is neither terminating nor recurring
⇒ 2 + 9∈I 11 = 3.316 624 79… is neither terminating nor recurring
⇒ 11 ∈ I 16 2 = 22.627 417… is neither terminating or recurring
⇒ 16 2 ∈ I 32 = 5.656 854 249… is neither terminating or recurring
⇒ 32 ∈ I 81 = 9 is a positive whole number ⇒ 81 ∈Z+ (and is
also rational) The answer is D.
Exercise 1B — Recurring decimals
1 a 18
= 0.125 (terminating)
b 12
= 0.5 (terminating)
c 13
= 0.333 3… = 0. 3&
d 619
= 0.315 789 473 684 210 526 315 789…
= 0.315 789 473 684 210 526 (as the number is rational and doesn’t terminate it must be recurring the pattern is beyond the range of the calculators)
e 117
= 0.058 823 529 410 588…
= 0.058 823 529 41 ( as the number is rational and doesn’t terminate it must be recurring the pattern is beyond the range of the calculators)
f 411
= 0.363 636… = 0. 36& &
g 59
= 0.555… = 0. 5&
h 716
= 0.437 5 (terminating)
i 925
= 0.36 (terminating)
j 57
= 0.714 285 714… = 0. 714 285
k 23
= 0.666… = 0. 6&
l 16
= 0.166 66… = 0.1 6&
m 34
= 0.75 (terminating)
n 313
= 0.230 769 230 7…
= 0. 230 769
o 521
= 0.238 095 238…
= 0. 238 095
p 231
= 0.064 516 129 032 258 064…
0.064 516 129 032 258= ( as the number is rational and doesn’t terminate it must be recurring the pattern is beyond the range of the calculators)
q 29
= 0.222 2… = 0. 2&
r 41333
= 0.123 123 123… = 0.123
s 58
= 0.625 (terminating)
t 1718
= 0.944 4… = 0.9 4&
u 817
= 0.470 588 235 294 117 647…
= 0.470 588 235 294 117 6 (as the number is rational and doesn’t terminate it must be recurring the pattern is beyond the range of the calculators)
v 723
= 0.304 347 826 086 956 521 739 130 434…
0.3043478260869565217391= (as the number is rational and doesn’t terminate it must be recurring the pattern is beyond the range of the calculators)
w 715
= 0.466 6… = 0.4 6&
x 322
= 0.136 363 6… = 0.1 36& &
y 733
= 0.212 121… = 0. 21& &
z 755
= 0.127 27… = 0.1 27& &
The recurring decimals are c, d, e, f, g, j, k, l, n, o, p, q, r, t, u, v, w, x, y. If a fraction can be written as a terminating decimal, the denominator will divide a power of 10, otherwise the fraction will be represented by a recurring decimal.
2 a x = 0.222 22 [1] 10x = 2.222 22 [2] Evaluating [2] − [1]: 10x − x = 2.222 2… −0.222 2… 9x = 2
x = 29
b x = 0.777 7 [1] 10x = 7.777 7 [2]
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M C 1 1 Q l d - 1 4 N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m
Evaluating [2] − [1]: 10x − x = 7.777 7… − 0.777 7… 9x = 7
x = 79
c x = 0.888 8 [1] 10x = 8.888 8 [2] Evaluating [2] − [1]: 10x − x = 8.888 8… − 0.888 8… 9x = 8
x = 89
d x = 0.555 5 [1] 10x = 5.555 5 [2] Evaluating [2] − [1]: 10x − x = 5.555 5… − 0.555 5… 9x = 5
x = 59
e x = 0.444 4 [1] 10x = 4.444 4 [2] Evaluating [2] − [1]: 10x − x = 4.444 4… − 0.444 4… 9x = 4
x = 49
f x = 0.166 666 [1] 10x = 1.666 6 [2] Evaluating [2] − [1]: 10x − x = 1.666 6… − 0.166 6… 9x = 1.5
x = 1.59
x = 1590
= 16
g x = 0.377 77 [1] 10x = 3.777 7 [2] Evaluating [2] − [1]: 10x − x = 3.777 7… − 0.377 7… 9x = 3.4
x = 3.49
x = 3490
= 1745
h x = 0.422 22 [1] 10x = 4.222 2 [2] Evaluating [2] − [1]: 10x − x = 4.222… − 0.422 2… 9x = 3.8
x = 3.89
x = 3890
= 1945
i x = 0.688 8 [1] 10x = 6.888 [2] Evaluating [2] − [1]: 10x − x = 6.888… − 0.688 8…
9x = 6.2
x = 6.29
x = 6290
= 3145
j x = 0.711 11 [1] 10x = 7.111 1 [2] Evaluating [2] − [1] 10x − x= 7.111… − 0.711… 9x = 6.4
x = 6.49
x = 6490
x = 3245
k x = 2.622 2 [1] 10x = 26.222 [2] Evaluating [2] − [1]: 10x − x = 26.222… − 2.622 2… 9x = 23.6
x = 23.69
x = 23690
= 11845
= 2 2845
l x = 0.535 3 [1] 100x = 53.535 3 [2] Evaluating [2] − [1]: 100x − x = 53.535 3… − 0.535 3… 99x = 53
x = 5399
m x = 0.121 2 [1] 100x = 12.121 2 [2] Evaluating [2] − [1]: 100x − x = 12.121 2…− 0.121 2… 99x = 12
x = 1299
= 433
n x = 1.343 4 [1] 100x = 134.343 4 [2] Evaluating [2] − [1] 100x − x = 134.343 4…− 1.343 4… 99x = 133
x = 13399
= 1 3499
o x = 3.741 41 [1] 100x = 374.141 4 [2] Evaluating [2] − [1] 100x − x = 374.141 4…− 3.741 41… 99x = 370.4
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N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m M C 1 1 Q l d - 1 5 x = 370.4
99
x = 3704990
= 1852495
= 3 367495
p x = 0.361 361 [1] 1000x = 361.361 361 [2] Evaluating [2] − [1] 1000x − x = 361.361 361…− 0.361 361… 999x = 361
x = 361999
q x = 0.427 427 [1] 1000x = 427.427 427 [2] Evaluating [2] − [1] 1000x − x = 427.427 427…− 0.427 427… 999x = 427
x = 427999
r x = 0.521 321 3 [1] 1000x = 521.321 321 [2] Evaluating [2] − [1] 1000x − x = 521.321 321…− 0.521 321 3… 999x = 520.8
x = 520.8999
x = 52089990
= 26044995
= 8681665
s x = 0.323 323 [1] 1000x = 323.323 323 [2] Evaluating [2] − [1] 1000x − x = 323.323 323…− 0.323 323… 999x = 323
x = 323999
t x = 3.456 456 [1] 1000x = 3456.456 456 [2] Evaluating [2] − [1] 1000x − x = 3456.456 456…− 3.456 456… 999x = 3453
x = 3453999
= 1151333
= 3 152333
u x = 0.722 2 [1] 10x = 7.222 2 [2] Evaluating [2] − [1] 10x − x = 7.222 2… − 0.722 2… 9x = 6.5
x = 6.59
x = 6590
x = 1318
v x = 0.523 33 [1] 10x = 5.233 3 [2] Evaluating [2] − [1] 10x − x = 5.233 3… − 0.523 33… 9x = 4.71
x = 4.719
x = 471900
x = 157300
w x = 0.624 747 [1] 100x = 62.474 7 [2] Evaluating [2] − [1] 100x − x = 62.474 7…− 0.624 747… 99x = 61.85
x = 61.8599
x = 61859900
= 12371980
x x = 0.623 444 [1] 10x = 6.234 44 [2] Evaluating [2] − [1] 10x − x = 6.234 44…− 0.623 444… 9x = 5.611
x = 5.6119
x = 56119000
y x = 0.153 846 153 846 [1] 1 000 000x = 153 846.153 846 [2] Evaluating [2] − [1] 1 000 000x − x = 153 846.153 846… − 0.153 846… 999 999x = 153 846
x = 153 846999 999
= 17 094111111
= 155410101
= 213
3 x = 0.787 8 [1] 100x = 78.787 8 [2] Evaluating [2] − [1] 100x − x = 78.787 8…− 0.787 8… 99x = 78
x = 7899
The answer is E. 4 x = 0.532 32 [1] 100x = 53.232 3 [2] Evaluating [2] − [1] 100x − x = 53.232 3…− 0.532 3… 99x = 52.7
x = 52.799
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M C 1 1 Q l d - 1 6 N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m
x = 527990
The answer is D. 5 12 = 0.5 (terminating decimal)
45 = 0.8 (terminating decimal)
37 = 0.428 571 428 5…
= 0. 428571 (recurring decimal) 13 = 0.333 3… = 0. 3& (recurring decimal)
1113
= 0.846 153 846…
= 0. 846153 (recurring decimal) The answer is C. 6 x = 0.369 369… [1] 1000x = 369.369 369… [2] Evaluating [2] − [1] 1000x − x = 369.369 369…− 0.369 369… 999x = 369
x = 369999
= 41111
The answer is E. 7 0.0 20 = 0.020 202 0… 0. 020 = 0.020 020… Therefore, Irene is correct and it can also be written as 0. 02 .
Exercise 1C — Surds 1 a 81 = 9∴ rational ⇒ not surd b 48 = 6.928 203 23… ∴ irrational ⇒ surd c 16 = 4∴ rational ⇒ not surd d 1.6 = 1.264 911 064… ∴ irrational ⇒ surd e 0.16 = 0.4 ∴ rational ⇒ not surd f 11 = 3.316 624 79… ∴ irrational ⇒ surd
g 34
= 0.866 025 403… ∴ irrational ⇒ surd
h 3 327
= 0.480 749 856… ∴ irrational ⇒ surd
i 1000 = 31.622 776 6… ∴ irrational ⇒ surd j 1.44 = 1.2 ∴ rational ⇒ not surd k 4 100 = 4 × 10 = 40 ∴ rational ⇒ not surd l 2 + 10 = 5.162 277 66… ∴ irrational ⇒ surd m 3 32 = 3.174 802 104… ∴ irrational ⇒ surd n 361 = 19 ∴ rational ⇒ not surd o 3 100 = 4.641 588 834… ∴ irrational ⇒ surd p 3 125 = 5 ∴ rational ⇒ not surd q 6 + 6 = 4.898 979 486… ∴ irrational ⇒ surd r 2π = 6.283 185 307… is irrational, but has no sign
⇒ not surd s 3 169 = 5.528 774 814… ∴ irrational ⇒ surd
t 78
= 0.935 414 346… ∴ irrational ⇒ surd
u 4 16 = 2∴ rational ⇒ not surd v ( 7 )2 = 7∴ rational ⇒ not surd w 3 33 = 3.207 534 33… irrational ⇒ surd x 0.0001 = 0.01 ∴ rational ⇒ not surd
y 5 32 = 2 ∴ rational ⇒ not surd
2 69
= 0.816 496 58…. ∴ irrational ⇒ surd
20 = 4.472 135 955… ∴ irrational ⇒ surd 54 = 7.348 469 228… ∴ irrational ⇒ surd 3 27 = 3 ∴ rational ⇒ not surd 9 = 3 ∴ rational ⇒ not surd The answer is A. 3 a Assume 3 is rational, that is it can be written in simplest
terms as 3 , 0a bb
= ≠ . As it is in simplest terms, a and b
have no common factors. Squaring both sides of the equation:
2
2
2 2
3
3 [1]
ab
b a
=
=
This means 3 is a factor of a2 and 3 will also be a factor of a. ∴ a = 3r
2 29 [2]a r= Substituting [1] in [2]:
2 2
2 2
3 9
3
b r
b r
=
=
This means 3 is a factor of b2 and 3 will also be a factor of b.
Therefore a and b both have a factor of 3. This contradicts the initial assumption that a and b will
have no common factors. ∴ 3 is not rational ∴ it must be irrational b Assume 5 is rational, that is it can be written in simplest
terms as 5 , 0a bb
= ≠ . As it is in simplest terms, a and b
have no common factors. Squaring both sides of the equation:
2
2
2 2
5
5 [1]
ab
b a
=
=
This means 5 is a factor of a2 and 5 will also be a factor of a. ∴ a = 5r
2 225 [2]a r= Substituting [1] in [2]:
2 2
2 2
5 25
5
b r
b r
=
=
This means 5 is a factor of b2 and 5 will also be a factor of b.
Therefore a and b both have a factor of 5. This contradicts the initial assumption that a and b will
have no common factors. ∴ 5 is not rational ∴ it must be irrational c Assume 7 is rational, that is it can be written in simplest
terms as 7 , 0a bb
= ≠ . As it is in simplest terms, a and b
have no common factors. Squaring both sides of the equation:
2
2
2 2
7
7 [1]
ab
b a
=
=
-
N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m M C 1 1 Q l d - 1 7 This means 7 is a factor of a2 and 7 will also be a factor of
a. ∴ a = 7r 2 249 [2]a r= Substituting [1] in [2]:
2 2
2 2
7 49
7
b r
b r
=
=
This means 7 is a factor of b2 and 7 will also be a factor of b.
Therefore a and b both have a factor of 7. This contradicts the initial assumption that a and b will
have no common factors. ∴ 7 is not rational ∴ it must be irrational
4 14
= 12
∴ rational ⇒ not surd
3 127
= 13
∴ rational ⇒ not surd
18
= 0.353 553 39… ∴ irrational ⇒ surd
21 = 4.582 575 695… ∴ irrational ⇒ surd 3 8 = 2 ∴ rational ⇒ not surd The answer is E. 5 π = 3.141 592 654…. is irrational, but has no root sign ⇒
not surd
149
= 17
is rational ⇒ not surd
12 = 3.464 101 615… is irrational ⇒ surd 16 = 4 is rational ⇒ not surd 3 + 1 = 2.732 05… is irrational ⇒ surd The answer is B. 6 6 7 , 7 6 , 9 2 , 18 are surds
14416
= 9 = 3 is an integer ∴ not a surd
25 = 5 is an integer. ∴ not a surd 6 7 = 15.874 507 87… is irrational ⇒ a surd 7 6 = 17.146 428 2… is irrational ⇒ a surd 9 2 = 12.727 922 06… is irrational ⇒ a surd 18 = 4.242 640 687… is irrational ⇒ a surd The answer is C. 7 If a is a multiple of 4, then a can be written as 4r. Let 6 a b= where b is a non-zero rational number. Substituting a = 4r:
6
6
2 6
4
4
2
r b
r b
r b
=
=
=
This means that 22 is a factor of b6. For b to be as small as possible, r = 24, making b = 2. (This occurs when a = 64.)
8 m = 1, 3 3
3
16 16 1
162.519 842 1...
m = ×
==
is irrational ⇒ surd m = 2, 3 3
3
16 16 2
323.174 802 104...
m = ×
==
is irrational ⇒ surd
m = 3, 3 3
3
16 16 3
483.634 241 186...
m = ×
==
is irrational ⇒ surd m = 4, 3 3
3
16 16 4
644
m = ×
==
= 3 16 4×
is rational ⇒ not a surd ∴ Smallest value of m so that 3 16m is not a surd is m = 4. Alternatively, let 3 16m p= where p is a positive integer:
3
3
2 3
16
16
4
m p
m p
m p
=
=
=
The smallest value of m that can satisfy this condition is m = 4.
Exercise 1D — Simplifying surds 1 a 12 4 3
4 3
2 3
= ×
= ×
=
b 18 9 2
9 2
3 2
= ×
= ×
=
c 24 4 6
4 6
2 6
= ×
= ×
=
d 56 4 14
4 14
2 14
= ×
= ×
=
e 27 9 3
9 3
3 3
= ×
= ×
=
f 75 25 3
25 3
5 3
= ×
= ×
=
g 125 25 5
25 5
5 5
= ×
= ×
=
h 99 9 11
9 11
3 11
= ×
= ×
=
i 54 9 6
9 6
3 6
= ×
= ×
=
j 60 4 15
4 15
2 15
= ×
= ×
=
k 112 16 7
16 7
4 7
= ×
= ×
=
-
M C 1 1 Q l d - 1 8 N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m
l 98 49 2
49 2
7 2
= ×
= ×
=
m 68 4 17
4 17
2 17
= ×
= ×
=
n 150 25 6
25 6
5 6
= ×
= ×
=
o 180 36 5
36 5
6 5
= ×
= ×
=
p 338 169 2
169 2
13 2
= ×
= ×
=
q 88 4 22
4 22
2 22
= ×
= ×
=
r 135 9 15
9 15
3 15
= ×
= ×
=
s 162 81 2
81 2
9 2
= ×
= ×
=
t 200 100 2
100 2
10 2
= ×
= ×
=
u 245 49 5
49 5
7 5
= ×
= ×
=
v 320 64 5
64 5
8 5
= ×
= ×
=
w 448 64 7
64 7
8 7
= ×
= ×
=
x 735 49 15
49 15
7 15
= ×
= ×
=
y 405 81 5
81 5
9 5
= ×
= ×
=
2 a 2 8 2 4 2
2 4 2
2 2 2
4 2
= ×
= ×
= ×
=
b 3 50 3 25 2
3 25 2
3 5 2
15 2
= ×
= ×
= ×
=
c 8 90 8 9 10
8 9 10
8 3 10
24 10
= ×
= ×
= ×
=
d 6 112 6 16 7
6 16 7
6 4 7
24 7
= ×
= ×
= ×
=
e 80 9 16 5
9 16 5
9 4 5
36 5
= ×
= ×
= ×
=
f 5 68 5 4 17
5 4 17
5 2 17
10 17
= ×
= ×
= ×
=
g 7 54 7 9 6
7 9 6
7 3 6
21 6
= ×
= ×
= ×
=
h 10 32 10 16 2
10 16 2
10 4 2
40 2
= ×
= ×
= ×
=
i 6 75 6 25 3
6 25 3
6 5 3
30 3
− = − ×
= − ×
= − ×
= −
j 3 252 3 36 7
3 36 7
3 6 7
18 7
= ×
= ×
= ×
=
k 7 80 7 16 5
7 16 5
7 4 5
28 5
− = − ×
= − ×
= − ×
= −
l 9 120 9 4 30
9 4 30
9 2 30
18 30
= ×
= ×
= ×
=
m 16 48 16 16 3
16 16 3
16 4 3
64 3
= ×
= ×
= ×
=
-
N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m M C 1 1 Q l d - 1 9 n 1 190 9 10
3 31 9 1031 3 103
10
= ×
= ×
= ×
=
o 1 1392 196 27 7
1 196 271 14 272 2
= ×
= ×
= ×
=
p 1 1625 255 5
5
= ×
=
q 1 1162 81 29 9
1 81 291 9 29
2
= ×
= ×
= ×
=
r 2 254 9 63 3
2 9 632 3 632 6
= ×
= ×
= ×
=
s 1 1192 64 34 4
1 64 341 8 342 3
= ×
= ×
= ×
=
t 1 1288 144 26 6
1 144 261 12 262 2
= ×
= ×
= ×
=
u 1 1135 9 159 9
1 9 1591 3 159
15 1 or 153 3
= ×
= ×
= ×
=
v 5 5320 64 52 2
5 64 52
= ×
= ×
5 8 5220 5
= ×
=
w 3 3175 25 710 10
3 25 7103 5 7
103 7 3or 7
2 2
= ×
= ×
= ×
=
x 7 7176 16 118 8
7 16 1187 4 1187 11 7or 11
2 2
= ×
= ×
= ×
=
y 4 4108 36 33 3
4 36 334 6 338 3
− = − ×
= − ×
= − ×
= −
3 a 2 216 164
a aa
= ×=
b 2 2 2 281 819
a b a bab
= × ×=
c 2 2
2
72 36 2
36 2
6 2
a a
a
a
= × ×
= ×
=
d 2 2 2 2
2 2
54 9 6
9 6
3 6
a b a b
a b
ab
= × × ×
= ×
=
e 2 2
2
90 9 10
9 10
3 10
a b a b
a b
a b
= × × ×
= ×
=
f 3 2
2
48 16 3
16 3
4 3
a b a a b
a ab
a ab
= × × × ×
= ×
=
g 4 4
4
2
338 169 2
169 2
13 2
a a
a
a
= × ×
= ×
=
h 4 2 4 2
4 2
2
150 25 6
25 6
5 6
a b a b
a b
a b
= × × ×
= ×
=
i 3 3 2 2
2 2
338 169 2
169 2
13 2
a b a a b b
a b ab
ab ab
= × × × × ×
= ×
=
-
M C 1 1 Q l d - 1 10 N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m
j 5 7 4 6
4 6
2 3
12 4 3
4 3
2 3
a b a a b b
a b ab
a b ab
= × × × × ×
= ×
=
k 3 5 2 4
2 4
2
68 4 17
4 17
2 17
a b a a b b
a b ab
ab ab
= × × × × ×
= ×
=
l 6 6
6
3
80 16 5
16 5
4 5
x y x y
x y
x y
= × × ×
= ×
=
m 6 4 6 4
6 4
3 2
125 25 5
25 5
5 5
x y x y
x y
x y
= × × ×
= ×
=
n 2 2 2
2 2
3 64 3 8
3 8
3 8
24
x y x y
x y
x y
x y
= × ×
= ×
= ×
=
o 3 2 2 2
2 2
5 80 5 16 5
5 16 5
5 4 5
20 5
x y x x y
x y x
xy x
xy x
= × × × ×
= ×
= ×
=
p 3 3 2 2
2 2
2 343 2 49 7
2 49 7
2 7 7
14 7
x y x x y y
x y xy
xy xy
xy xy
= × × × × ×
= ×
= ×
=
q 7 5 6 4
6 4
3 2
3 2
6 162 6 81 2
6 81 2
6 9 2
54 2
c d c c d d
c d cd
c d cd
c d cd
= × × × × ×
= ×
= ×
=
r 4 5 4 4
4 4
2 2
2 2
3 126 3 9 14
3 9 14
3 3 14
9 14
c d c d d
c d d
c d d
c d d
= × × × ×
= ×
= ×
=
s 7 9 6 8
6 8
3 4
3 4
2 405 2 81 5
2 81 5
2 9 5
18 5
c d c c d d
c d cd
c d cd
c d cd
= × × × × ×
= ×
= ×
=
t 10 10 10 10
10 10
5 5
5 5
4 294 4 49 6
4 49 6
4 7 6
28 6
c d c d
c d
c d
c d
= × × ×
= ×
= ×
=
u 1 188 4 222 2
1 4 222
ef e f
ef
= × × ×
= ×
1 2 222
22
ef
ef
= ×
=
v 4 6 4 6
4 6
2 3
2 3
1 1120 4 303 3
1 4 3031 2 3032 303
e f e f
e f
e f
e f
= × × ×
= ×
= ×
=
w 11 11 10 10
10 10
5 5
5 5
1 1392 196 22 2
1 196 221 14 227 2
e f e e f f
e f ef
e f ef
e f ef
= × × × × ×
= ×
= ×
=
x 12 5 12 4
12 4
6 2
6 2
3 3175 25 720 20
3 25 7203 5 7203 74
e f e f f
e f f
e f f
e f f
= × × × ×
= ×
= ×
=
y 3 9 2 8
2 8
4
4
1 154 9 627 27
1 9 6271 3 6271 69
x y x x y y
x y xy
xy xy
xy xy
= × × × × ×
= ×
= ×
=
z 10 12 10 12
10 12
5 6
5 6
1 1108 36 318 18
1 36 3181 6 3
181 33
x y x y
x y
x y
x y
= × × ×
= ×
= ×
=
4 45 9 5
3 5
= ×
=
The answer is E. 5 3 128 3 64 2
3 64 2
3 8 2
24 2
= ×
= ×
= ×
=
The answer is C.
6 1 1539 49 117 7
1 49 117
= ×
= ×
1 7 117
11
= ×
=
The answer is D.
-
N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m M C 1 1 Q l d - 1 11 7 4 3 4 2
4 2
2
213
1 1325 25 1315 15
1 25 13151 5 13
1513
x y x y y
x y y
x y y
x y y
− = − × × × ×
= − ×
= − ×
= −
The answer is C.
Exercise 1E — Addition and subtraction of surds 1 a 3 5 + 4 5 = (3 + 4) 5 = 7 5 b 6 2 + 11 2 = (6 + 11) 2 = 17 2 c 2 3 + 5 3 + 3 = (2 + 5 + 1) 3 = 8 3 d 6 7 + 8 7 + 5 7 = (6 + 8 + 5) 7 = 19 7 e 8 5 + 3 3 + 7 5 + 2 3 = (8 + 7) 5 + (3 + 2) 3 = 15 5 + 5 3 f 2 6 + 9 2 + 6 2 + 5 6 = (2 + 5) 6 + (9 + 6) 2 = 7 6 + 15 2 g 6 11 − 2 11 = (6 − 2) 11 = 4 11 h 12 13 − 5 13 − 2 13 = (12 − 5 − 2) 13 = 5 13 i 7 2 + 9 2 − 3 2 = (7 + 9 − 3) 2 = 13 2 j 3 7 − 2 5 + 7 7 − 9 5 = (3 + 7) 7 − (2 + 9) 5 = 10 7 − 11 5 k 9 6 + 12 6 − 17 6 − 7 6 = (9 + 12 − 17 − 7) 6 = −3 6 l 5 2 − 12 2 − 3 6 + 8 6 = (5 − 12) 2 + (− 3 + 8) 6 = −7 2 + 5 6 m 12 3 − 8 7 + 5 3 − 10 7 = (12 + 5) 3 − (8 + 10) 7 = 17 3 − 18 7 n xy + 7 xy − 3 xy
= (1 + 7 − 3) xy
= 5 xy
o 2 x + 5 y + 6 x − 2 y
= (2 + 6) x + (5 − 2) y
= 8 x + 3 y
p 3 x + 4 y + 7 xy − 2 x − 9 y
= (3 − 2) x + (4 − 9) y + 7 xy
= x − 5 y + 7 xy
2 a 200 − 300 = 100 × 2 − 100 × 3 = 10 2 − 10 3 = 10 ( 2 − 3 ) b 18 + 50 − 72 = 9 2× + 25 2× − 36 2× = 9 × 2 + 25 × 2 − 36 × 2 = 3 2 + 5 2 − 6 2 = 2 2 c 125 − 150 + 600 = 25 5× − 25 6× + 100 6× = 25 × 5 − 25 × 6 + 100 × 6 = 5 5 − 5 6 + 10 6 = 5 5 + 5 6 = 5 ( 5 + 6 ) d 96 − 5 24 + 12 = 16 6× − 5 4 6× + 4 3× = 16 × 6 − 5 × 4 × 6 + 4 × 3 = 4 6 − 5 × 2 6 + 2 3 = 4 6 − 10 6 + 2 3 = −6 6 + 2 3 e 27 − 3 + 75 = 9 3× − 3 + 25 3× = 9 × 3 − 3 + 25 × 3 = 3 3 − 3 + 5 3 = 7 3 f 8 + 18 + 50 = 4 2× + 9 2× + 25 2× = 4 × 2 + 9 × 2 + 25 × 2 = 2 2 + 3 2 + 5 2 = 10 2 g 2 20 − 3 5 + 45 = 2 4 5× − 3 5 + 9 5× = 2 × 2 5 − 3 5 + 3 5 = 4 5 − 3 5 + 3 5 = 4 5 h 45 + 20 = 9 5× + 4 5× = 9 × 5 + 4 × 5 = 3 5 + 2 5 = 5 5 i 6 12 3 27 7 3 18
6 4 3 3 9 3 7 3 9 2
6 4 3 3 9 3 7 3 3 2
6 2 3 3 3 3 7 3 3 2
12 3 9 3 7 3 3 2
14 3 3 2
+ − +
= × + × − + ×
= × + × − +
= × + × − +
= + − +
= +
-
M C 1 1 Q l d - 1 12 N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m
j 44 99 121 3 11
4 11 9 11 11 3 11
4 11 9 11 11 3 11
2 11 3 11 11 3 11
4 11 11
− + −
= × − × + −
= × − × + −
= − + −
= − +
k 150 24 96 108
25 6 4 6 16 6 36 3
25 6 4 6 16 6 36 3
5 6 2 6 4 6 6 3
3 6 6 3
+ − +
= × + × − × + ×
= × + × − × + ×
= + − +
= +
l 98 − 2 50 + 5 32 = 49 2× − 2 25 2× + 5 16 2× = 49 × 2 − 2 25 × 2 + 5 16 × 2 = 7 2 − 2 × 5 2 + 5 × 4 2 = 7 2 − 10 2 + 20 2 = 17 2 m 3 90 5 60 3 40 100
3 9 10 5 4 15 3 4 10 10
3 9 10 5 4 15 3 4 10 10
3 3 10 5 2 15 3 2 10 10
9 10 10 15 6 10 10
15 10 10 15 10
− + +
= × − × + × +
= × − × + × +
= × − × + × +
= − + +
= − +
n 2 99 − 44 − 176 = 2 9 11× − 4 11× − 16 11× = 2 9 × 11 − 4 × 11 − 16 × 11 = 2 × 3 11 − 2 11 − 4 11 = 6 11 − 2 11 − 4 11 = 0 o 5 11 + 7 44 − 9 99 + 2 121 = 5 11 + 7 4 11× − 9 9 11× + 2 × 11 = 5 11 + 7 × 4 × 11 − 9 × 9 × 11 + 22 = 5 11 + 7 × 2 11 − 9 × 3 11 + 22 = 5 11 + 14 11 − 27 11 + 22 = −8 11 + 22 p 5 3 + 8 27 − 4 3 + 2 147 = 5 3 + 8 9 3× − 4 3 + 2 49 3× = 5 3 + 8 × 3 3 − 4 3 + 2× 7 3 = 5 3 + 24 3 − 4 3 + 14 3 = 39 3 q 2 30 + 5 120 + 60 − 6 135 = 2 30 + 5 4 30× + 4 15× − 6 9 15× = 2 30 + 5 × 2 30 + 2 15 − 6× 3 15 = 2 30 + 10 30 + 2 15 − 18 15 = 12 30 − 16 15 r 20 50 80 120 60
4 5 25 2 16 5 4 30 4 15
2 5 5 2 4 5 2 30 2 15
2 5 5 2 2 30 2 15
− − − +
= × − × − × − × + ×
= − − − +
= − − − +
s 6 ab − 12ab + 2 9ab + 3 27ab = 6 ab − 4 3ab× + 2 9 ab× + 3 9 3ab×
= 6 ab − 2 3ab + 2 × 3 ab + 3 × 3 3ab = 6 ab − 2 3ab + 6 ab + 9 3ab = 12 ab + 7 3ab
t 1 505
+ 2 987
− 3 324
= 1 25 25
× + 2 49 27
× − 3 16 24
×
= 15
× 5 2 + 27
× 7 2 − 34
× 4 2
= 2 + 2 2 − 3 2 = 0
u 1 982
+ 1 483
+ 1 123
= 1 49 22
× + 1 16 33
× + 1 4 33
×
= 12
× 7 2 + 13
× 4 3 + 13
× 2 3
= 7 22
+ 4 33
+ 2 33
= 7 22
+ 2 3
v 1 51216
− 5 1288
+ 1 726
= 1 256 216
× − 5 64 28
× + 1 36 26
×
= 116
× 16 2 − 58
× 8 2 + 16
× 6 2
= 2 − 5 2 + 2 = −3 2
w 1 328
− 7 186
+ 3 72
= 1 16 28
× − 7 9 26
× + 3 36 2×
= 18
× 4 2 − 76
× 3 2 + 3 × 6 2
= 1 22
− 7 22
+ 18 2
= 15 2
x 1 278
+ 7 1216
− 5 4832
= 18
9 3× + 7 4 316
× − 5 16 332
×
= 18
× 3 3 + 716
× 2 3 − 532
× 4 3
= 3 38
+ 7 38
− 5 38
= 5 38
3 a 7 a − 8a + 9 9a − 32a = 7 a − 4 2a× + 9 × 3 a − 16 2a× = 7 a − 2 2a + 27 a − 4 2a = 34 a − 6 2a b 10 a − 15 27a + 8 12a + 14 9a = 10 a − 15 9 3a× + 8 4 3a× + 14 × 9 a× = 10 a − 15 × 3 3a + 8 × 2 3a + 14 × 3 a = 10 a − 45 3a + 16 3a + 42 a = 52 a − 29 3a
-
N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m M C 1 1 Q l d - 1 13 c 150ab + 96ab − 54ab = 25 6ab× + 16 6ab× − 9 6ab× = 25 × 6ab + 16 × 6ab − 9 × 6ab = 5 6ab + 4 6ab − 3 6ab = 6 6ab
d 16 24a − 24a + 4 28a + 96a
= 16 × 2a − 4 6a× + 4 24 2a× + 16 6a× = 32a − 2 6a + 4 × 2a 2 + 4 6a = 32a + 2 6a + 8a 2
e 3 3 3
2 2 2
8 72 98
4 2 36 2 49 2
2 2 6 2 7 2
2
a a a
a a a a a a
a a a a a a
a a
+ −
= × × × + × × × − × × ×
= + −
=
f 1 362
a + 1 1284
a − 1 1446
a
= 12
× 6 a + 1 64 24
a× − 16
× 12 a
= 3 a + 14
× 8 2a − 2 a
= 3 a + 2 2a − 2 a = a + 2 2a
g 39a + 53a
= 29a a× + 43a a× = 3a a + a2 3a
h 6 5a b + 3a b − 5 5a b
= 6 4a a b× × + 2a a b× × − 5 4a a b× × = 6a2 ab + a ab − 5a2 ab = a2 ab + a ab = (a2 + a) ab
i ab ab + 3ab 2a b + 3 39a b
= ab ab + 3ab 2a b× + 2 29a a b b× × × = ab ab + 3ab × a b + 3ab ab = 4ab ab + 3a2b b
j 3a b + 5 ab − 2 ab + 5 3a b
= 2a a b× × + 5 ab − 2 ab + 5 2a a b× × = a ab + 3 ab + 5a ab = 6a ab + 3 ab = (6a + 3) ab = 3(2a + 1) ab
k 3 2 5 6
2 2 4 6
2 3
2 3
2 3
32 5 8 48
16 2 5 4 2 16 3
4 2 5 2 2 4 3
4 2 10 2 4 3
6 2 4 3
a b ab a a b
a a b ab a a a b
ab a ab a a b a
ab a ab a a b a
ab a a b a
− +
= × × × − × + × × ×
= − × +
= − +
= − +
l 24a b + 5 2a b − 3 29a b
= 24a b× + 5 2a b× − 3 29a b× = 2a b + 5a b − 3 × 3a b = 2a b + 5a b − 9a b = −2a b
4 112 − 63 = 16 7× − 9 7× = 4 7 − 3 7 = 7 The answer is D.
5 2 40a − 6 272ab
= 2 4 10a× − 6 236 2b a× = 2 × 2 10a − 6 × 6b 2a = 4 10a − 36b 2a The answer is E.
6 27 10010
a − 22 255
a + 21 726
b
= 27 10010
a − 22 255
a + 21 36 26
b ×
= 710
× 10a − 25
× 5a + 16
× 6b 2
= 7a − 2a + b 2 = 5a + b 2 The answer is A.
7 3 6243a b − 27a
= 2 681 3a b a× − 9 3a× = 9ab3 3a − 3 3a = 3a (9ab3 − 3) = 3 3a (3ab3 − 1) The answer is E.
8 2 2150c d − cd 96 − c 254d
= 2 225 6c d × − cd 16 6× − c 29 6d × = 5cd 6 − cd × 4 6 − c × 3d 6 = 5cd 6 − 4cd 6 − 3cd 6 = −2cd 6 The answer is B. 9 a P = 4L = 4 × 18 = 4 × 9 2× = 4 × 3 2 = 12 2 cm b P = 48 + 6 + 27 + 54 + 24 + 3 = 16 3× + 6 + 9 3× + 9 6× + 4 6× + 3 = 4 3 + 6 + 3 3 + 3 6 + 2 6 + 3 = (8 3 + 6 6 ) cm c P = 2( 5 + 2) + 2(7 − 3 ) = 2 5 + 4 + 14 − 2 3 = (2 5 + 18 − 2 3 ) cm d P = π × 45 = π × 9 5×
= π × 3 5 = 3π 5 m e P = 2(5 2 − 5 ) + 4( 5 + 2 2 ) = 10 2 − 2 5 + 4 5 + 8 2 = (18 2 + 2 5 ) m f P = 3 44 − 99 + 2 44 + 4 44 + 2 99 = 9 44 + 99
-
M C 1 1 Q l d - 1 14 N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m
= 9 4 11× + 9 11× = 9 × 2 11 + 3 11 = 18 11 + 3 11 = 21 11 m
Exercise 1F — Multiplication of surds 1 a 2 7 2 7
14
× = ×
=
b 5 11 5 11
55
× = ×
=
c 6 7 6 7
42
× = ×
=
d 2 12 2 4 3
2 2 3
2 2 3
2 6
× = × ×
= ×
= ×
=
e 8 6 4 2 6
2 2 6
2 2 6
2 12
2 4 3
2 2 3
4 3
× = × ×
= ×
= ×
=
= ×
= ×
=
f 12 6 12 6
72
36 2
6 2
× = ×
=
= ×
=
g 10 10 10 10
10010
× = ×
==
h 5 75 5 25 3
5 5 3
5 5 3
5 15
× = × ×
= ×
= ×
=
i 21 3 21 3
63
9 7
3 7
× = ×
=
= ×
=
j 2 8 5 2 4 2 5
2 2 2 5
4 2 5
4 10
× = × ×
= × ×
= ×
=
k 27 3 3 9 3 3 3
3 3 3 3
9 3 3
9 99 327
× = × ×
= ×
= ×
== ×=
l 45 60 9 5 4 15
3 5 2 15
6 5 15
6 75
6 25 3
6 5 3
30 3
× = × × ×
= ×
= ×
=
= ×
= ×
=
m 5 3 2 11 5 2 3 11
10 33
× = × ×
=
n 6 2 4 48 6 2 4 16 3
6 2 4 4 3
96 2 3
96 6
× = × ×
= × ×
= ×
=
o 10 15 6 3 10 6 15 3
60 45
60 9 5
60 3 5
180 5
× = × ×
=
= ×
= ×
=
p 9 2 7 2 9 7 2 2
63 463 2126
× = × ×
== ×=
q 4 20 3 5 4 4 5 3 5
4 2 5 3 5
24 5 5
24 2524 5120
× = × ×
= × ×
= ×
== ×=
r 6 18 2 8 6 9 2 2 4 2
6 3 2 2 2 2
72 2 2
72 472 2144
× = × × ×
= × × ×
= ×
== ×=
s 10 6 3 8 10 6 3 4 2
10 6 3 2 2
60 6 2
60 12
× = × ×
= × ×
= ×
=
60 4 3
60 2 3
120 3
= ×
= ×
=
t 9 20 4 15 9 4 5 4 15
9 2 5 4 15
72 5 15
72 75
72 25 3
72 5 3
360 3
× = × ×
= × ×
= ×
=
= ×
= ×
=
-
N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m M C 1 1 Q l d - 1 15 u 1 148 2 2 16 3 2 2
4 41 4 3 2 242 3 2
2 6
× = × ×
= × ×
= ×
=
v 1 1 1 172 3 36 2 32 3 2 3
1 6 2 36
2 3
6
× = × × ×
= × ×
= ×
=
w 1 148 2 3 16 3 2 39 9
× = × ×
1 4 3 2 398 3 398 998 3983
22 3
= × ×
= ×
=
= ×
=
=
x 2 1 2 14 125 2 25 53 5 3 5
4 5 5154 5 4 or 5
3 3
× = × × ×
= ×
=
y 1 1 1 160 40 4 15 4 1010 5 10 5
1 12 15 2 1010 5
× = × × ×
= × × ×
2 15 10252 150252 25 6252 5 6252 6 2 or 6
5 5
= ×
=
= ×
= ×
=
z 3 2 330 10 30 104 5 10
3 300103 100 3
103 10 3
103 3
× = ×
=
= ×
= ×
=
2 a 3 2
2 2
xy x y
xy x x y
×
= × × ×
2
2
xy xy x
xy xy x
xy x y
xy x y
x y y
= ×
= ×
=
= ×
=
b 3 4 2 2
2 4
2
2 3
x y x y
x x y xy
xy x xy
x y x
×
= × × ×
= ×
=
c 4 2 5 3
4 2 4 2
2 2
2 2
4 2
4 2
4 2
4 2
3 6
3 6
3 6
3 6
18
9 2
3 2
3 2
a b a b
a b a a b b
a b a b ab
a b a b ab
a b ab
a b ab
a b ab
a b ab
×
= × × × × × × ×
= ×
= × ×
=
= ×
= ×
=
d 2 3 5
2 2 4
2
2
2
2
2
5 10
5 10
5 10
50
25 2
5 2
5 2
a b ac
a b b a c c
ab b c ac
abc abc
abc abc
abc abc
abc abc
×
= × × × × × × ×
= ×
=
= ×
= ×
=
e 7 3 4
6 2 4
3 2
4 2 2
4 2 2
4 2
5 2
12 6
4 3 6
2 3 6
2 (18 )
2 9 2
2 3 2
6 2
a b a b
a a b a a b
a ab ab a
a b a b
a b a b
a b a b
a b b
×
= × × × × × × × ×
= ×
=
= × × ×
= ×
=
f 4 3 2 5
4 2 2 4
2 2
18 2
9 2 2
3 2 2
a b a b
a b b a b b
a b b ab b
×
= × × × × × × × ×
= ×
3 3 2
3 3
3 4
3 4
3 2
6
a b b
a b b
a b
=
= ×
=
g 3 2 2 3
2 2 2 2
2 2
2 2
2 2
2 2
15 6
15 6
15 6
90
9 10
3 10
3 10
x y x y
x x y x y y
xy x xy y
x y xy
x y xy
x y xy
x y xy
×
= × × × × × × ×
= ×
=
= ×
= ×
=
-
M C 1 1 Q l d - 1 16 N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m
h 7 5 3
6 4 2
3 2
5 2 2
5 2 2
5
6 2
3 10 5
3 10 5
3 10 5
3 50
3 25 2
3 5 2
15 2
x y x y
x x y x x y y
x xy x y xy
x y x y
x y x y
x y xy
x y
×
= × × × × × × × ×
= ×
=
= × × ×
= ×
=
i 3 3 2 61 15 3 32
a b a b×
2 2 3
3
2 4
2 4
2 4
2 4
2 4
1 15 3 321 15 3 323 15 323 4523 9 523 3 529 52
a a b b ab
ab ab ab
a b ab
a b ab
a b ab
a b ab
a b ab
= × × × × × ×
= × ×
= ×
=
= ×
= ×
=
j 4 2 3 31 112 63 4
a b a b×
4 2 2 2
2
3 2
3 2
3 2
3 2
3 2
1 14 3 63 41 12 3 63 41 3 661 1861 9 261 3 261 22
a b a a b b
a b ab ab
a b ab
a b ab
a b ab
a b ab
a b ab
= × × × × × ×
= × ×
= ×
=
= ×
= ×
=
3 a
( )( )
2
2
22
2
7 2
7 2
49 2
98 cm
A s=
=
= ×
= ×
=
b
( )( )
2
2
22
2
5 3
5 3
25 3
75 cm
A rπ
π
π
ππ
=
=
= × ×
= ×
=
c
2
5 11 2 4
5 11 2 2
20 11 m
A l w= ×
= ×
= × ×
=
d 1212
12
12
2
2 8 3 3
2 4 2 3 3
2 2 2 3 3
6 6 m
A b h= ×
= × ×
= × × ×
= × × ×
=
e The shape can be thought of as a rectangle and a circle (the 2 ends will combine to make 1 circle).
( )( )
2
2
22
2
6 56 5 8 82
6 5 8 4 2 3 5
6 5 8 2 2 3 5
96 10 9 5
96 10 45 m
A π
π
π
ππ
⎛ ⎞= × + ⎜ ⎟⎜ ⎟
⎝ ⎠
= × × +
= × × + × ×
= + × ×
= +
f Divide the shape into 2 rectangles; one will be 5 10 m × 6 6 m and the other will be 2 10 m × 3 6 m.
2
5 10 6 6 2 10 3 6
30 60 6 60
36 60
36 4 15
36 2 15
72 15 m
A = × + ×
= +
=
= ×
= ×
=
4 3 30 5 6 15 30 6
15 5 6 6
15 6 5
90 5
× = ×
= × ×
= ×
=
The answer is E.
5 5 2 6 3
4 2 6 2
2 3
5 2
8 5
4 2 5
2 2 5
2 10
x y x y
x x y x y y
x y x x y y
x y xy
×
= × × × × × ×
= ×
=
The solution is C.
6 7 2 4 3
6 2 4 2
3 2
5 2
3 18 2
3 18 23 18 23
16
x y x y
x x y x y y
x y x x y y
x y xy
×
= × × × ×
= ×
=
The solution is D. 7 Area of a triangle is 1
2base × perpendicular height.
1 4 6 5 32
A = × ×
10 18
10 9 2
=
= ×
-
N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m M C 1 1 Q l d - 1 17
2
10 3 2
30 2 m
= ×
=
The solution is A. 8 Area of the base is side2 Volume is 1
3 area of base × height
( )( )( )( )
2
22
2
22
3
1 12 8 20 831 12 4 2 20 4 231 144 2 2 20 2 23
48 2 2 40 2
48 4 2 40 2
15360 2 units
V = ×
= × × × ×
= × × ×
= × ×
= × × ×
=
Exercise 1G — The Distributive Law 1 a ( )3 7 6
3 7 3 6
21 6 3
+
= × + ×
= +
b ( )( )
5 18 7
5 3 2 7
5 3 2 7 5
3 10 7 5
−
= −
= × −
= −
c ( )5 2 25 2 5 2
2 5 10
−
= × − ×
= −
d ( )2 3 52 3 2 5
6 10
+
= × + ×
= +
e ( )( )( )
7 3 72 12
7 3 6 2 2 3
7 18 2 2 3
7 18 2 7 2 3
126 2 14 3
−
= × −
= −
= × − ×
= −
f ( )6 5 14 46 5 14 6 4
5 84 4 6
5 2 21 4 6
10 21 4 6
−
= × − ×
= −
= × −
= −
g ( )( )( )
2 2 6 18 7 15
2 2 6 3 2 7 15
2 2 18 2 7 15
2 2 18 2 2 2 7 15
36 4 14 30
36 2 14 30
72 14 30
+
= × +
= +
= × + ×
= +
= × +
= +
h ( )( )
( )
5 12 3 5 4 8
5 2 3 3 5 4 2 2
10 3 3 5 8 2
10 3 3 5 10 3 8 2
30 15 80 6
− −
= − × − ×
= − −
= − × − × −
= − +
i ( )2 3 4 6 2 32 3 4 6 2 3 2 3
8 18 4 9
8 3 2 4 3
24 2 12
− −
= − × − × −
= − +
= − × + ×
= − +
2 a ( )( )18 5 5 318 5 18 3 5 5 5 3
90 3 18 5 5 15
3 10 3 3 2 5 5 15
3 10 9 2 5 5 15
− +
= × + × − × − ×
= + − −
= + × − −
= + − −
b ( )( )7 5 2 5 3 77 2 5 7 3 7 5 2 5 5 3 7
2 35 3 49 2 25 3 35
35 3 7 2 5
35 21 10
35 11
+ −
= × − × + × − ×
= − + −
= − − × + ×
= − − +
= − −
c ( )( )( )( )( )( )
4 8 2 6 8 3 6
4 2 2 2 6 2 2 3 6
8 2 2 6 2 2 3 6
8 2 2 2 8 2 3 6 2 6 2 2 2 6 3 6
16 4 24 12 4 12 6 36
16 2 20 12 6 6
32 20 2 3 36
4 40 3
+ −
= × + −
= + −
= × + × − + × + × −
= − + −
= × − − ×
= − × −
= − −
d ( )( )3 6 2 5 4 2 3 20− −
( )( )( )( )3 6 2 5 4 2 3 2 5
3 6 2 5 4 2 6 5
3 6 4 2 3 6 6 5 2 5 4 2 2 5 6 5
12 12 18 30 8 10 12 25
12 2 3 18 30 8 10 12 5
24 3 18 30 8 10 60
= − − ×
= − −
= × − × − × − × −
= − − +
= × − − + ×
= − − +
e ( )( )7 8 6 3 4 2 5 6+ −
( )( )( )( )7 2 2 6 3 4 2 5 6
14 2 6 3 4 2 5 6
14 2 4 2 14 2 5 6 6 3 4 2 6 3 5 6
56 4 70 12 24 6 30 18
56 2 70 2 3 24 6 30 3 2
112 140 3 24 6 90 2
= × + −
= + −
= × − × + × − ×
= − + −
= × − × + − ×
= − + −
-
M C 1 1 Q l d - 1 18 N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m
f ( )( )11 2 3 2 5 8− −
( )( )11 2 3 2 5 2 211 2 5 11 2 2 2 3 2 5 2 3 2 2
2 55 2 22 4 15 4 6
= − −
= × − × − × − × −
= − − +
g ( )( )2 7 3 2 5 5 7 22 7 5 5 2 7 7 2 3 2 5 5 3 2 7 2
10 35 14 14 15 10 21 4
10 35 14 14 15 10 21 2
10 35 14 14 15 10 42
− +
= × + × − × − ×
= + − −
= + − − ×
= + − −
h ( )( )( )( )( )( )
5 18 3 3 2 18 6
5 3 2 3 3 2 3 2 6
15 2 3 3 6 2 6
15 2 6 2 15 2 6 3 3 6 2 3 3 6
90 4 15 12 18 6 3 18
90 2 15 2 3 18 6 3 3 2
180 30 3 18 6 9 2
− −
= × − × −
= − −
= × − × − × − × −
= − − +
= × − × − + ×
= − − +
i ( )( )
2 2
5 2 3 4
5 3 5 4 2 3 2 4
15 20 6 8
15 26 8
x y x y
x x x y y x y y
x xy xy y
x xy y
+ +
= × + + + × + ×
= + + +
= + +
j ( )( )8 10 2 10x y x y− + ( )( )
( ) ( )
2 2 10 2 10
2 2 2 2 2 10 10 2 10 10
2 22 2 2 20 20 10
2 2 2 2 5 2 5 10
4 4 5 2 5 10
4 2 5 10
x y x y
x x x y y x y y
x xy xy y
x xy xy y
x xy xy y
x xy y
= − +
= × + × − × − ×
= + − −
= × + × − −
= + − −
= + −
3 a ( )( )
2
2 2
2 5
2 2 2 5 5
2 10 2 25
27 10 2
+
= + × +
= + +
= +
b ( )( ) ( )
2
2 2
6 10
6 2 6 10 10
6 2 60 10
16 2 2 15
16 4 15
+
= + × +
= + +
= + ×
= +
c ( )( ) ( )
2
2 2
3 15
3 2 3 15 15
3 2 45 15
18 2 3 5
18 6 5
+
= + × +
= + +
= + ×
= +
d ( )( ) ( )
2
2 2
3 5 2
3 2 3 5 2 5 2
3 10 6 25 2
3 10 6 50
53 10 6
+
= + × +
= + + ×
= + +
= +
e ( )( ) ( )
2
2 2
8 3 3
8 2 8 3 3 3 3
8 6 24 9 3
8 6 2 6 27
35 12 6
+
= + × +
= + + ×
= + × +
= +
f ( )( ) ( )
2
2 2
2 2 3 5
2 2 2 2 2 3 5 3 5
4 2 12 10 9 5
8 12 10 45
53 12 10
+
= + × × +
= × + + ×
= + +
= +
g ( )( ) ( )
2
2 2
3 6 5 2
3 6 2 3 6 5 2 5 2
9 6 30 12 25 2
54 30 2 3 50
104 60 3
+
= + × × +
= × + + ×
= + × +
= +
h ( )( ) ( )
2
2 2
5 3
5 2 5 3 3
5 6 5 9
14 6 5
−
= + × − + −
= − +
= −
i ( )( ) ( )
2
2 2
7 3
7 2 7 3 3
7 2 21 3
10 2 21
−
= + × − + −
= − +
= −
j ( )( ) ( )
2
2 2
2 8 5
2 8 2 2 8 5 5
4 8 4 40 5
32 4 2 10 5
37 8 10
−
= + × × − + −
= × − +
= − × +
= −
4 a ( )( ) ( )2 23 7 3 7 3 73 49
46
+ − = −
= −= −
b ( )( ) ( ) ( )2 219 1 19 1 19 119 118
+ − = −
= −=
c ( )( ) ( )2 22 5 3 2 5 3 2 5 34 5 9
+ − = −
= × −
-
N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m M C 1 1 Q l d - 1 19 20 9
11= −=
d ( )( ) ( )2 23 11 7 3 11 7 3 11 79 11 4999 4950
+ − = −
= × −= −=
e ( )( ) ( ) ( )2 28 2 8 2 8 28 26
+ − = −
= −=
f ( )( ) ( ) ( )2 210 12 10 12 10 1210 12
2
+ − = −
= −= −
g ( )( ) ( ) ( )2 213 3 13 3 13 313 310
− + = −
= −=
h ( )( ) ( ) ( )2 27 12 7 12 7 127 12
5
− + = −
= −= −
i ( )( ) ( ) ( )22 3 5 2 3 5 2 3 54 3 512 57
− + = −
= × −= −=
j ( )( ) ( ) ( )2 23 7 12 3 7 12 3 7 129 7 1263 1251
+ − = −
= × −= −=
k ( )( ) ( ) ( )2 22 10 14 2 10 14 2 10 144 10 1440 1426
+ − = −
= × −= −=
l ( )( ) ( ) ( )2 218 19 18 19 18 1918 19
1
− + = −
= −= −
m ( )( ) ( ) ( )2 213 6 13 6 13 613 67
− + = −
= −=
n ( )( ) ( ) ( )2 23 5 2 7 3 5 2 7 3 5 2 79 5 4 745 2817
+ − = −
= × − ×= −=
o ( )( ) ( ) ( )2 26 3 3 5 6 3 3 5 6 3 3 536 3 9 5108 4563
− + = −
= × − ×= −=
p ( )( ) ( ) ( )2 25 2 6 5 2 6 5 2 625 2 650 644
− + = −
= × −= −=
q ( )( ) ( ) ( )2 27 2 3 5 7 2 3 5 7 2 3 549 2 9 598 4553
− + = −
= × − ×= −=
r ( )( ) ( ) ( )2 211 3 2 5 11 3 2 5 11 3 2 5121 3 4 5363 20343
+ − = −
= × − ×= −=
s ( )( ) ( ) ( )2 26 3 2 8 6 3 2 8 6 3 2 836 3 4 8108 3276
+ − = −
= × − ×= −=
t ( )( ) ( ) ( )2 27 2 3 9 7 2 3 9 7 2 3 949 2 9 998 8117
− + = −
= × − ×= −=
u ( )( ) ( ) ( )2 2x y x y x yx y
− + = −
= −
v ( )( ) ( ) ( )2 22 3 2 3 2 32 3
x y x y x y
x y
− + = −
= −
w ( )( ) ( ) ( )2 23 4 3 4 3 49 16
x y x y x y
x y
− + = −
= −
x ( )( )( ) ( )2 2
2
3
2 5 2 5
2 5
4 25
4 25
x x y x x y
x x y
x x y
x y
+ −
= −
= × −
= −
y ( )( )( ) ( )2 2
2 2
7 3 7 3
7 3
49 9
x y y x x y y x
x y y x
x y y x
− +
= −
= −
z ( )( )2 2 2 29 5 9 5x y xy x y xy− + ( ) ( )
2 22 2
2 2
9 5
81 25
x y xy
x y xy
= −
= −
5 ( )15 5 3 15 5 15 375 45
25 3 9 5
5 3 3 5
− = × − ×
= −
= × − ×
= −
The solution is A (C is not in simplest form). 6 ( )( )5 8 2 7 6 5 3 3
5 8 6 5 5 8 3 3 2 7 6 5 2 7 3 3
30 40 15 24 12 35 6 21
+ −
= × + × − + × + −
= − + −
30 2 10 15 2 6 12 35 6 21
60 10 30 6 12 35 6 21
= × − × + −
= − + −
The solution is C.
-
M C 1 1 Q l d - 1 20 N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m
7 ( ) ( ) ( )2 2 27 5 2 3 7 5 2 7 5 2 3 2 349 5 28 15 4 3
245 28 15 12
257 28 15
− = + × × − + −
= × − + ×
= − +
= −
The solution is E.
8 ( )( )( ) ( )
2 2
2 22
2
15 4 15 4
15 4
225 16
x y xy x y xy
x y xy
x y xy
+ −
= −
= −
The solution is D.
9 a ( )( ) ( )
22
2 2
3 5 2 3
3 5 2 3 5 2 3 2 3
9 5 12 15 4 3
45 12 15 12
57 12 15
x = −
= + × × − + −
= × − + ×
= − +
= −
b ( ) ( )2 3 2 57 12 15 3 3 5 2 3 257 12 15 9 5 6 3 2
59 12 15 9 5 6 3
x x+ + = − + − +
= − + − +
= − + −
Exercise 1H — Division of surds
1 a 15 1533
5
=
=
b 14 1422
7
=
=
c 8 8224
2
=
==
d 72 7266
12
2 3
=
=
=
e 60 6010106
=
=
f 90 9066
15
=
=
g 128 12888
164
=
==
h 45 45125125925
35
=
=
=
i 18 1 184 64 61 34
34
=
=
=
j 2 24 2 243 33 32 832 2 234 2
3
=
=
= ×
=
k 65 1 652 132 131 52
52
=
=
=
l 5 72 7251212
5 6
=
=
m 96 9688
12
2 3
=
=
=
n 2 63 2 635 75 72 952 3565115
=
=
= ×
=
=
o 7 44 7 4414 1114 117 4
147 2
1414141
=
=
= ×
=
=
p 336 3361414
24
2 6
=
=
=
q 9 63 9 6315 715 73 953 35
=
=
= ×
-
N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m M C 1 1 Q l d - 1 21 9
5415
=
=
r 540 5402020
27
3 3
=
=
=
s 2040 20403030
68
2 17
=
=
=
t 12 99 12 9915 1115 114 954 35125225
=
=
= ×
=
=
u 4 3 2
22 5
x y x y yxy yx yxy
=
=
v 6 11 3 5
6 712 15
3 21
x y x y yx y yx y
x y
=
=
w 7 97 9
6 8
3 4
16 1688
2
2
xy xyx yx y
x y
x y
=
=
=
x 4 3 4 3
22
3
72 7222
36
6
x y x yxyxy
x y
x xy
=
=
=
2 a 8 12 8 12
5 7 2 3 5 2 7 3
9 13
7 10
2 3
12 12
12
12
2 3
xy x y x x y y
x y x y x x y y
x yx y
x y
xy y
× × × ×× =
× × ×
=
=
=
b 2 3 7 2 9 5
5 74 4 3
4
2
6 3 188127 3
29
x y x y x yx yx y xy
xy
× =
=
2
2
23
23
xy
xy
=
=
c 2 4 9 3 11 7
10 73 6 7
2 2 10 2 203 55 32 432 234
3
a b a b a ba ba b a b
a
a
a
× =
=
= ×
=
d 5 7 3 5 7 3
56 2 6 2 5
5 7 3
6 5 2
8 8
11 3
5
3
3 6 3 622 2 2
3 6
2 2
184
9 24
ab a b ab a ba ba b a b a b
a b a b
a a b b
a ba b
ba
× = ×
× × × × ×=× × × × ×
=
× ×=×
4
2
2
3 22
3 22
b b
a ab b
a a
× ×=×
=
e 3 4 6 3 3
5 2 3 5 2 4 6
2 3 2 8
6 8 6 3
mn m n mn mn
m n mn m n m n÷ = ×
2 6
9 8
7 2
7 2
3
3
1618
894 2
92 2
32 2
3
m nm n
m n
m n
mm n
m n m
=
=
×=
=
=
f 3 5 8
3 2 5
5 3 2
2 6 6
m n m n
m n mn÷
3 5
3 2 5 8
3 5
3 2 5 8
4 6
8 10
4 4
2 2
2 2
5 3 6
2 6 2
5 6 3
2 6 2
30 3
2 1215 11 415 12
152
m n mn
m n m n
m n m n
m n m n
m n
m n
m n
m n
m n
= ×
× × × × ×=× × × × ×
=
= ×
×=
=
-
M C 1 1 Q l d - 1 22 N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m
3 75 7555
15
=
=
The answer is B.
4 9 18 9 1821 321 33 673 6
7
=
= ×
=
The answer is E.
5 5 8 4 8
3 2 2 2
2 4
2 4
3
3
10 10
20 4 5
102 5
102 5
22
22
x y x x y
x y x x y
x y xxy x
x y xxy x
xy
xy
× × ×=
× × × ×
=
=
=
=
The answer is A.
6 4 7 3 11 4
3 5 4 6
5 2
2 3
3
2 6 129 4 36
2 36
33
x y x y x yxy x y x y
x y xx y
x xy
× =
=
=
The answer is C. 7 a
28 39 7 3
28 397 3
28 397 3
4 13 m
A l w
w
w
w
= ×
= ×
=
=
=
b
12 30 3 5
12 303 5
12 303 5
4 6 cm
A bh
b
b
b
=
= ×
=
=
=
c 12121 55 6 523 5
21 553 5
21 113
7 11 m
A bh
h
h
h
h
=
= × ×
= ×
=
=
=
d V = lwh
90 21 5 2 3 6
15 12
90 2115 1290 2115 12
764
762
3 7 m
h
h
h
h
= × ×
= ×
=
=
=
= ×
=
e V = π r2h
( )( )
2
22
315 13 3 7
3 7
9 763
315 1363
315 1363
5 13 cm
h
h
hh
h
h
π π
π
ππ
ππ
ππ
=
= × × ×
= × × ×=
=
=
=
f V = 13
area of base × h
160 75 24 1538 15
60 758 15
60 758 15
15 52
15 5 cm2
h
h
h
h
π π
ππππ
π
= × ×
= ×
=
=
=
=
8 a 2 802
80
16 5
4 5 m/s
v ×=
=
= ×
=
b 2 25060
50060253
5 m/s3
v ×=
=
=
=
c 2 480120
8
4 2
2 2 m/s
v ×=
=
= ×
=
-
N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m M C 1 1 Q l d - 1 23 9 a V = 84 L
= 84 000 cm3 when 23
full.
If H is the height when 23
full, then
84 000 20 3 30 6
600 18
600 3 2
1 800 284 000
1 800 2140 cm3 2
V lwH
H
H
H
H
H
H
=
= × ×
= ×
= × ×
= ×
=
=
Therefore 23
h = 1403 2
, where h
is height when full.
3 1402 3 270
270 2
2 270 2
235 2 cm
h = ×
=
= ×
=
=
The height of the tank is 35 2 cm.
b 23
V = 84,
where V is volume when full.
V = 84 × 32
= 126 The full capacity of the tank is 126 litres.
Exercise 1I — Rationalising denominators
1 a 5 5 22 2 2
5 22
= ×
=
b 7 7 33 3 3
7 33
= ×
=
c 4 4 1111 11 11
4 1111
= ×
=
d 8 8 66 6 6
8 66
4 63
= ×
=
=
e 12 2 37 7
2 3 77 7
2 217
=
= ×
=
f 15 15 66 6 6
9069 10
63 10
6102
= ×
=
×=
=
=
g 2 3 2 3 55 5 5
2 155
= ×
=
h 3 7 3 7 55 5 5
3 355
= ×
=
i 5 2 5 2 32 3 2 3 3
5 62 35 6
6
= ×
=×
=
j 4 3 4 3 53 5 3 5 5
4 153 54 15
15
= ×
=×
=
k 5 14 5 147 8 7 2 2
5 14 214 2 25 2814 25 4 714 2
5 2 728
5 714
=×
= ×
=×
×=×
×=
=
l 16 3 16 3 56 5 6 5 5
16 156 5
= ×
=×
-
M C 1 1 Q l d - 1 24 N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m
16 1530
8 1515
=
=
m 8 3 8 3 77 7 7 7 7
8 217 78 21
49
= ×
=×
=
n 8 60 8 2 1528 2 7
8 15 77 7
8 1057
×=
= ×
=
o 2 35 2 35 3 143 14 3 14 3 14
6 4909 14
6 7 10126
103
= ×
=×
×=
=
2 a 6 12 6 2 3 33 3 3
+ += ×
6 3 2 3 33 3
18 2 33
3 2 63
3( 2 2)3
2 2
× + ×=×
+ ×=
+=
+=
= +
b 15 22 15 22 66 6 6
15 6 22 66
90 1326
3 10 2 336
− −= ×
× − ×=
−=
−=
c 6 2 15 6 2 15 1010 10 10
6 2 10 15 1010
6 20 15010
6 2 5 25 610
12 5 5 610
− −= ×
× − ×=
−=
× − ×=
−=
d 2 18 3 2 2 9 2 3 25 5
2 3 2 3 25
6 2 3 25
9 2 55 5
9 105
+ × × +=
× +=
+=
= ×
=
e 3 5 6 7 3 5 6 7 28 2 2 2
3 5 2 6 7 22 2
3 10 6 144
+ += ×
× + ×=×
+=
f 4 2 3 8 4 2 3 2 22 3 2 3
4 2 6 22 3
10 2 32 3 3
10 62 35 6
3
+ + ×=
+=
= ×
=×
=
g 3 11 4 5 3 11 4 5 218 3 2 2
3 11 2 4 5 23 2
3 22 4 106
− −= ×
× − ×=×
−=
h 2 7 2 5 2 7 2 5 312 2 3 3
2 7 3 2 5 32 3
2 21 2 156
2( 21 15)6
21 153
− −= ×
× − ×=×
−=
−=
−=
i
( )
7 12 5 6 7 2 3 5 6 36 3 6 3 3
14 3 5 186 3
42 5 3 218
3 14 5 2
3 614 5 2
6
− × −= ×
× −=×
− ×=
−=
×−=
-
N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m M C 1 1 Q l d - 1 25 j 6 2 5 6 2 5 2
4 8 4 2 2 26 2 10
8 212 10
16
− −= ××
× −=×
−=
k 6 3 5 5 6 3 5 5 57 20 7 2 5 5
6 15 5 514 5
6 15 2570
− −= ××
− ×=×−=
3 12 12 33 3 3
12 33
4 3
= ×
=
=
The solution is B.
4 8 5 8 59 12 9 2 3
8 5 318 3 38 1518 38 15
544 15
27
=×
= ×
=×
=
=
The answer is D.
5 7 5 6 7 7 5 6 7 312 2 3 3
7 15 6 212 3
7 15 6 216
− −= ×
−=×−=
The solution is C.
6 5 5 3 3 5 5 3 3 28 8 8 2 2 2
5 10 3 616 2
5 10 3 632
− −= ××
−=×−=
The solution is A.
7 a x2 = 2 37
373 77 7217
x
x
=
= ±
= ± ×
= ±
b 2
2
3 553
x
x
=
=
535 33 3153
x = ±
= ± ×
= ±
c 2
2
2
6 4 12
6 1683
83
2 2 33 3
2 63
x
x
x
x
− =
=
=
= ±
= ± ×
= ±
Exercise 1J — Rationalising denominators using conjugate surds
1 a
2 2
1 1 5 25 2 5 2 5 2
1 5 1 2( 5) 2
5 25 45 21
5 2
−= ×+ + −
× + × −=−
−=−−=
= −
b
( )22
1 1 3 63 6 3 6 3 6
1 3 1 6
3 6
3 69 6
3 63
+= ×− − +
× + ×=−
+=−
+=
c
( ) ( )2 2
1 1 8 58 5 8 5 8 5
1 8 1 5
8 5
8 58 5
2 2 53
+= ×− − +
× + ×=−
+=−
+=
d
2 2
1 1 2 6 72 6 7 2 6 7 2 6 7
1 2 6 1 7(2 6) ( 7)
2 6 74 6 7
+= ×− − +
× + ×=−
+=× −
-
M C 1 1 Q l d - 1 26 N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m
2 6 724 7
2 6 717
+=−+=
e
2 2
4 4 2 11 132 11 13 2 11 13 2 11 13
4 2 11 4 13(2 11) ( 13)
8 11 4 134 11 13
8 11 4 1344 13
8 11 4 1331
+= ×− − +
× + ×=−
+=× −
+=−+=
f
2 2
7 7 2 12 2 52 12 2 5 2 12 2 5 2 12 2 5
7 2 12 7 2 5(2 12) (2 5)
−= ×+ + −
× + × −=−
( )( )
7 2 2 3 2 354 12 4 5
2 2 21 35
2 24 10
2 21 3514
× × −=× − ×
−=
−
−=
g
2 2
5 3 5 3 3 5 4 23 5 4 2 3 5 4 2 3 5 4 2
5 3 3 5 5 3 4 2(3 5) (4 2)
15 15 20 69 5 16 2
15 15 20 645 32
15 15 20 613
−= ×+ + −
× + × −=−
−=× − ×
−=−−=
h 9 32 33 12−
( ) ( )
( )( )
2 2
9 3 2 33 122 33 12 2 33 129 3 2 33 9 3 12
2 33 12
18 99 9 364 33 12
18 3 11 9 6132 12
6 3 3 11 9
6 22 2
9 11 920
+= ×− +
× + ×=−
+=× −
× + ×=−
× +=
−
+=
i 2 55 7 20
−−
( ) ( )2 2
2 5 5 7 205 7 20 5 7 20
2 5 7 2 20 5 5 7 5 20
5 7 20
5 14 40 25 7 5 2 525 7 20
5 14 2 10 25 7 10 5175 20
5 14 2 10 25 7 10 5155
− += ×− +
× + × − × − ×=−
+ − − ×=× −
+ − −=−
+ − −=
j 8 38 3
−+
( )
( )
( )
2
3
8 3 8 38 3 8 3
8 3
8 98 2 8 3 3
18 2 2 2 3 9
− −×+ −
−=
−+ × − + −
=−
= − − × × +
( )17 12 212 2 17
= − −
= −
k 12 712 7
−+
( )
( ) ( )
2
2 2
12 7 12 712 7 12 7
12 7
12 7
12 2 12 7 7
512 2 2 3 7 7
519 4 21
5
− −= ×+ −
−=
−
+ × − + −=
− × × +=
−=
l 11 722 14
+−
2 2
11 7 22 1422 14 22 1411 22 11 14 7 22 7 14
( 22) ( 14)
242 154 154 9822 14
121 2 2 154 49 28
11 2 2 154 7 28
18 2 2 1548
9 2 1544
+ += ×− +
× + × + × + ×=−
+ + +=−
× + + ×=
+ +=
+=
+=
-
N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m M C 1 1 Q l d - 1 27 m 5 3
4 10 3 18−+
( ) ( )2 2
5 3 4 10 3 184 10 3 18 4 10 3 18
5 4 10 5 3 18 3 4 10 3 3 18
4 10 3 18
4 50 5 3 3 2 4 30 3 3 3 216 10 9 18
4 5 2 9 10 4 30 9 6160 162
20 2 9 10 4 30 9 62
20 2 9 10 4 30 9 62
− −= ×+ −
× + × − − × − × −=−
− × × − + × ×=× − ×
× − − +=−
− − +=−
− + + −=
n 2 8 3 23 24 2 6
−−
2 2 2 3 23 2 6 2 64 2 3 26 6 2 6
× −=× −
−=−
2 64 6 6
124 62 3243
12
= ×
=×
=
=
o 3 6 2 124 18 3 8
++
3 6 2 2 34 3 2 3 2 23 6 4 3
12 2 6 23 6 4 3 2
18 2 23 12 4 6
18 23 2 3 4 6
363 3 2 6
18
+ ×=× + ×
+=+
+= ×
+=×
× +=
+=
p 5 2 3 32 6 3 12
+−
( ) ( )2 2
5 2 3 3 2 6 3 122 6 3 12 2 6 3 125 2 2 6 5 2 3 12 3 3 2 6 3 3 3 12
2 6 3 12
10 12 15 24 6 18 9 364 6 9 12
10 2 3 15 2 6 6 3 2 9 624 108
+ += ×− +× + × + × + ×=
−
+ + +=× − ×
× + × + × + ×=−
( )
( )
20 3 30 6 18 2 5484
2 10 3 15 6 9 2 27
2 4210 3 15 6 9 2 27
42
+ + +=−
+ + +=
− ×− + + +
=
q 4 5 106 15 20
++
( ) ( )2 2
4 5 10 6 15 206 15 20 6 15 204 5 6 15 4 5 20 10 6 15 10 20
6 15 20
24 75 4 100 6 150 20036 15 20
24 5 3 4 10 6 5 6 10 2540 20
120 3 40 30 6 10 2520
+ −= ×+ −
× + × − + × + × −=−
− + −=× −
× − × + × −=−
− + −=
( )10 12 3 4 3 6 210 52
12 3 4 3 6 252
− + −=
×− + −=
r 4 15 2 32 30 5 2
−−
( ) ( )
( )
2 2
4 15 2 3 2 30 5 22 30 5 2 2 30 5 24 15 2 30 4 15 5 2 2 3 2 30 2 3 5 2
2 30 5 2
8 450 20 30 4 90 10 64 30 25 2
8 15 2 20 30 4 3 10 10 6120 50
120 2 20 30 12 10 10 670
2 60 2 10 30 6 10 5 6
2 3560 2 10 30 6 10 5 6
35
− += ×− +× + × − × − ×=
−
+ − −=× − ×
× + − × −=−
+ − −=
+ − −=
×+ − −=
s 2 7 5 35 7 3 3
+−
2 2
2 7 5 3 5 7 3 35 7 3 3 5 7 3 32 7 5 7 2 7 3 3 5 3 5 7 5 3 3 3
(5 7) (3 3)
10 7 6 21 25 21 15 325 7 9 3
70 31 21 45175 27
115 31 21148
+ += ×− +× + × + × + ×=
−
× + + + ×=× − ×
+ +=−
+=
-
M C 1 1 Q l d - 1 28 N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m
t 2 11 3 32 11 3 3
−+
2
22
2 2
2 11 3 3 2 11 3 32 11 3 3 2 11 3 3
(2 11 3 3)(2 11 3 3)(2 11 3 3)
(2 11) 2 2 11 3 3 3 3(2 11) (3 3)
4 11 12 33 9 34 11 9 3
44 12 33 2744 27
71 12 3317
− −= ×+ −
−=+ −
− × × + −=−
× − + ×=× − ×
− +=−
−=
u 4 12 3 83 6 5 2
−−
4 2 3 3 2 23 6 5 2
× − ×=−
( )
2 2
8 3 6 2 3 6 5 23 6 5 2 3 6 5 28 3 3 6 8 3 5 2 6 2 3 6 6 2 5 2
(3 6) (5 2)
24 18 40 6 18 12 30 29 6 25 2
24 9 2 40 6 18 4 3 6054 50
24 3 2 40 6 18 2 3 604
72 2 40 6 36 3 604
4 18 2 10 6 9 3 15
418 2 10 6 9 3 15
− += ×− +× + × − × − ×=
−
+ − − ×=× − ×
× + − × −=−
× + − × −=
+ − −=
+ − −=
= + − −
v 3 8 6 37 2 3
+−
2 2
3 2 2 6 37 2 3
6 2 6 3 7 2 37 2 3 7 2 3
6 2 7 2 6 2 3 6 3 7 2 6 3 3(7 2) ( 3)
42 2 6 6 42 6 6 349 2 3
84 48 6 1898 3
102 48 695
× +=−
+ += ×− +
× + × + × + ×=−
× + + + ×=× −
+ +=−
+=
w 3 11 2 73 14 4 11
−+
3 11 2 7 3 14 4 113 14 4 11 3 14 4 11
− −= ×+ +
3 11 3 14 3 11 4 11 2 7 3 14 2 7 4 112 2(3 14) (4 11)
9 154 12 11 6 98 8 779 14 16 11
9 154 132 6 49 2 8 77126 176
9 154 132 6 7 2 8 7750
9 154 132 42 2 8 7750
9 154 132 42 2 8 7750
× + × − − × − × −=−
− × − × +=× − ×
− − × +=−
− − × +=−
− − +=−
− + + −=
x 4 15 2 53 5 15
+−
2 2
4 15 2 5 3 5 153 5 15 3 5 15
4 15 3 5 4 15 15 2 5 3 5 2 5 15(3 5) ( 15)
+ += ×− +× + × + × + ×=
−
12 75 4 15 6 5 2 759 5 15
+ × + × +=× −
14 75 60 3045 15
14 25 3 9030
14 5 3 9030
70 3 9030
10(7 3 9)30
7 3 93
+ +=−
× +=
× +=
+=
+=
+=
y 3 7 5 235 2 2
−+
( ) ( )2 2
3 7 5 2 35 2 235 2 2 35 2 2
3 7 35 3 7 2 2 5 2 35 5 2 2 2
35 2 2
3 245 6 14 5 70 10 235 4 2
3 7 5 6 14 5 70 2035 8
21 5 6 14 5 70 2027
− −= ×+ −× + × − − × − × −=
−
− − + ×=− ×
× − − +=−
− − +=
z 3 6 156 2 3
−+
( ) ( )2 2
3 6 15 6 2 36 2 3 6 2 3
3 6 6 3 6 2 3 15 6 15 2 3
6 2 3
3 6 6 18 90 2 456 4 3
− −= ×+ −× + × − − × − × −=
−
× − − +=− ×
-
N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m M C 1 1 Q l d - 1 29
( )
18 6 3 2 3 10 2 3 56 12
18 18 2 3 10 6 56
3 6 6 2 10 2 5
3 26 6 2 10 2 5
2
− × − + ×=−
− − +=−
− − + + −=
− ×− + + −=
2 a 1 18 2 2 8 2
a = +− −
( )
( )
( )
2 2
1 18 2 2 2 2
1 2 12 12 2 1
2 1
2 2 1
2 12 2 1
2 12
=− −
+= ×+−
+=⎛ ⎞−⎜ ⎟⎝ ⎠
+=−
+=
( )
( )
( )
2 2
1 12 8 2 2 2 2 2
1 2 2 12 2 12 2 2 1
2 2 1
2 2 2 1
2 2 12 4 2 1
2 2 114
=− × −
+= ×+−
+=⎛ ⎞−⎜ ⎟⎝ ⎠
+=× −
+=
2 1 2 2 12 14
2 1 7 2 2 12 7 14
7 2 7 2 2 114 14
9 2 814
a + += +
+ += × +
+ += +
+=
b 1 12 7 2 3 3 7 3
b = −+ +
( )
( ) ( )
( )
2 2
1 1 7 32 7 2 3 7 32 7 3
7 3
2 7 3
7 32 7 3
7 38
−= ×+ −+
−=⎛ ⎞−⎜ ⎟⎝ ⎠
−=−
−=
( ) ( )2 2
1 1 3 7 33 7 3 3 7 3 3 7 3
3 7 3
3 7 3
3 7 39 7 3
3 7 360
−= ×+ + −
−=−
−=× −
−=
7 3 3 7 38 60
7 3 15 3 7 3 28 15 60 2
15 7 15 3 6 7 2 3120 120
15 7 15 3 6 7 2 3120
9 7 13 3120
b − −= −
− −= × − ×
− −= −
− − +=
−=
c 3 7 4 83 5 3 5 3 3
×− +
2 2
3 7 8 23 5 3 5 3 3
24 143 5 5 3 5 3 3 3 5 3 3 3
24 143 5 9 15 15 3 3
24 1415 8 15 9
24 146 8 15
24 142(3 4 15)
12 14 3 4 153 4 15 3 4 1536 14 48 210
3 (4 15)
36 14 48 2109 16 15
36 14 48 210231
36 14 48 210231
16 210 12 1477
= ×− +
=× + × − × − ×
=× + − − ×
=+ −
=+
=+
−= ×+ −
−=−
−=− ×
−=−
− +=
−=
d 2 3 4 6 36 2 3 2 6 3 3
+×− +
8 18 2 36 2 6 6 3 3 2 3 2 6 2 3 3 3
8 9 2 62 6 3 18 4 18 6 38 3 2 6
12 18 18
+ ×=× + × − × − ×
× +=× + − − ×× +=− −
-
M C 1 1 Q l d - 1 30 N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m
( )
( )
2 2
24 2 66 3 2
3(8 2 2)3(2 2)
8 2 2 2 22 2 2 2
8 2 2 8 2 2 2 2 2 2(2 ) ( 2)
16 2 16 4 2 24 2
12 14 22
2 6 7 2
26 7 2
+=− −
+=− +
− + −= ×+ −
− × + × − × +=−
− + − +=−
−=
−=
= −
e 3 5 7 2 27 2 5 2
−÷+ +
( )
2 2
3 5 5 27 2 7 2 2
3 5 3 107 7 7 2 2 2 7 2 2 2
15 3 107 2 14 14 415 3 10 3 14
3 14 3 1415 3 15 14 3 10 3 3 10 14
3 ( 14)
45 15 14 9 10 3 1409 14
45 15 14 9 10 3 4 355
45 15 14 9 10 6 355
45 15 14 9 10 6 35
5
+= ×+ −
× +=× + × − + × + × −
+=− + −+ += ×− +× + × + × + ×=
−
+ + +=−
+ + + ×=−
+ + +=−
+ + += −
f 2 2 3 2 2 32 2 3 12 2 6 3
+ +÷− +
2 2
2 2 3 12 2 6 32 2 3 2 2 32 2 12 2 2 2 6 3 3 12 2 3 6 3
(2 2) ( 3)
24 2 12 6 12 6 6 38 3
48 24 6 185
66 24 65
+ += ×− +× + × + × + ×=
−
× + + + ×=−
+ +=
+=
g 7 8 2 8 2 73 7 3 8 3 8 3 7
g + −= +− +
( )
7 8 7 2 2 7 2 23 7 3 8 7 2 23 7 2 2
+ + += ×− +−
( ) ( )
( )
( )
( )
2 2
7 7 7 2 2 2 2 7 2 2 2 2
3 7 2 2
7 2 14 2 14 4 23 7 4 2
7 4 14 83 1
15 4 143
15 4 14
3
× + × + × + ×=⎛ ⎞−⎜ ⎟⎝ ⎠
+ + + ×=− ×
+ +=−
+=−
− +=
( )
( ) ( )2 2
2 8 2 73 8 3 7
2 2 2 2 7 2 2 72 2 73 2 2 7
4 2 2 2 4 2 7 2 7 2 2 2 7 7
3 2 2 7
−+
× − −= ×−+
× + × − − × − × −=⎛ ⎞−⎜ ⎟⎝ ⎠
( )
( )
8 2 4 14 4 14 2 73 4 2 7
16 8 14 143 1
30 8 143
× − − + ×=× −
− +=
−=
( )
( )
15 4 14 30 8 143 3
15 4 14 30 8 143
15 12 143
3 5 4 14
35 4 14
g− + −= +
− − + −=
−=
−=
= −
h 3 7 2 5 7 22 7 11 7 2 11
h + −= +− +
2 2
3 7 2 3 7 2 2 7 112 7 11 2 7 11 2 7 11
3 7 2 7 3 7 11 2 2 7 2 11(2 7) ( 11)
6 7 3 77 4 7 2 114 7 11
42 3 77 4 7 2 1128 11
42 3 77 4 7 2 1117
+ + += ×− − +
× + × + × + ×=−
× + + +=× −
+ + +=−
+ + +=
2 2
5 7 2 5 7 2 7 2 117 2 11 7 2 11 7 2 11
5 7 7 5 7 2 11 2 7 2 2 11( 7) (2 11)
− − −= ×+ + −
× + × − − × − × −=−
-
N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m M C 1 1 Q l d - 1 31
5 7 10 77 2 7 4 117 4 11
35 10 77 2 7 4 117 44
35 10 77 2 7 4 1137
35 10 77 2 7 4 1137
× − − +=− ×
− − +=−
− − +=−
− + + −=
42 3 77 4 7 2 1117
35 10 77 2 7 4 1137
42 3 77 4 7 2 11 3717 37
35 10 77 2 7 4 11 1737 17
1554 111 77 148 7 74 11629
595 170 77 34 7 68 11629
959 281 77 182 7 6 11629
h + + +=
− + + −+
+ + += ×
− + + −+ ×
+ + +=
− + + −+
+ + +=
i 13 5 11 213 5 11 2
i + += −− −
( )( ) ( )
( )
2
2 2
13 5 13 5 13 513 5 13 5 13 5
13 5
13 5
13 2 13 5 513 5
18 2 658
2 9 65
89 65
4
+ + += ×− − +
+=
−
+ × +=−
+=
+=
+=
( )( )
2
2 2
11 2 11 2 11 211 2 11 2 11 2
11 2
11 2
11 2 11 2 411 4
15 4 117
+ + += ×− − +
+=
−
+ × +=−
+=
9 65 15 4 114 7
9 65 7 15 4 11 44 7 7 4
63 7 65 60 16 1128 28
i + += −
+ += × − ×
+ += −
63 7 65 60 16 1128
3 7 65 16 1128
+ − −=
+ −=
j 5 6 2 6 2 54 5 4 6 3 6 3 5
j + −= −− −
( )( )
( ) ( )
( )
( )( )
2
2 2
5 6 5 6 5 64 5 4 6 5 64 5 6
5 6
4 5 6
5 2 5 6 64 5 6
11 2 304 1
11 2 30
4
+ + += ×− +−
+=
⎛ ⎞−⎜ ⎟⎝ ⎠
+ × +=−
+=−
− +=
( )( )
2 6 52 6 2 53 6 3 5 3 6 5
23
−− =− −
=
( )
( )
11 2 30 24 3
11 2 30 3 2 44 3 3 4
33 6 30 812 12
41 6 3012
41 6 30
12
j− +
= −
− −= × − ×
− −= −
− −=
− +=
3 a 2 3 53 2 3 2 5
+ −+ +
( )( )( )
( )( ) ( ) ( )( ) ( )
( ) ( )
( )
2 2
3 2 3 2 52 3 53 2 3 2 5 3 2 3 2 5
2 3 3 2 3 2 3 2 5 5 3 2 3 5 2 5
3 2 3 2 5
6 4 3 3 3 2 3 4 5 2 15 3 5 2 15 2 5
9 12 3 4 3 4 56 6 10 4 3 3 3 4 5 3 5 2 15 2 15
9 12 20 12 322 7 3 7 5 4 15
1 12 322 7 3 7 5 4 15 1 12 3
1 12 3 1 12 3
22 1 12 3 7
+ −+ −= ×+ + + −
+ + − + − + + ×=
+ −
+ + + × − + − + + ×=
+ + × − ×+ + + + − − − −=
+ − ++ − −=
++ − − −= ×
+ −
− +=
( ) ( ) ( )3 1 12 3 7 5 1 12 3 4 15 1 12 31 144 3
− − − − −
− ×
-
M C 1 1 Q l d - 1 32 N u m b e r s y s t e m s : t h e R e a l N u m b e r S y s t e m
2
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