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

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 +

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]

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

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

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