INDICES AND SURDS (BILANGAN BERPANGKAT DAN BENTUK …€¦ · (BILANGAN BERPANGKAT DAN BENTUK AKAR)...

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INDICES AND SURDS (BILANGAN BERPANGKAT DAN BENTUK AKAR)

Where n is a positive integer, na is defined as:

na a x a x a x ..... x a

n faktor

where a is called the base, and n, the index or exponent or power.

For example, 45 5 x 5 x 5 x 5

We shall restrict ourselves to positive bases (a > 0). Extending the definition to zero, negative and fractional

indices, we have the following results:

For a > 0 and positive integers p and q:

p qp

q

pp

p

po aaaaa

aa ,,1

,1

1

For example, 5 35

3

44

1

3

3 7755,8

1

2

12,12 ando

With these extended definitions, the following rules of indices hold for positive base, a, and any rational

indices, m and n.

m n m n

mm n

n

nm m n

a x a aa

aruler for same basea

a a

n n n

n n

n

a.b a x bruler for same index

a a

b b


A number that cannot be expressed as a fraction of two integers is called an irrational number. Some

examples of irrational numbers are ,7,2 3 , etc. An irrational number involving a root is called a surd.

General rules involving surds:

n n np a q a p q . a ; n n n

p a q a p q a

n n na . b a.b ;

n

n

n

a a

bb

OVERVIEW

LAWS OF INDICES LAWS OF SURDS

a. am x an = am+n

b. am : an = am-n

c. (am)n = amn

d. a0 = 1

e. m

n mna a

f. a-n = n

1

a

g. an x bn = (ab)n

h. am : bm =

ma

b

i. m

n

n m

1a

a

j.

n na b

b a

k.

m m

n na b

b a

a. x . x x

b. x . y xy

c. a a x

xx

d. xyx x

ory yy

e. a x b x (a b) x

f. a x b x (a b) x

g. x y x y x y

h. 2 2a x b y a x b y a x b y

i. a x ya

x yx y

j. 2

x y x 2 xy y

k. 2

x y x 2 xy y


Exercise:

1. Express each of the following in the surd form.

a. 3

1

x b. 4

5

6 c. 7

3

yx d. 3

5x y

e. 2

15

7

f.

21

32x . y

g.

1 1

3 4x . y

h. 3

5

3

12

.

yx i.

315

74x . y

2. Evaluate the following without the use of calculator:

a.

31

3 22

3

2

4 x 2

2

b.

1

1 32

2 4

2

5 x 5

3

c.

31

2 24

1

2

4 x 9

1

4

d.

1

1 52

3 6

1

2

5 x 5

4

e.

32

3

f. 2n 1 1 n n 1

9 x 3 : 27

3. If x y y z z x

a 0 , a a a

is equal ….

4. Simplify :

a.

1 1

2 2

x y

x y

b.

n 1 n 2

n 1 n 2

3 3

3 3

c.

n 1 2n 3

n 2n 1

4 2

4 2

5. If x 1

3 2

, then 2x

3

= ….

6. If x y

3 7 and 7 3 , find x . y

7. Express each of the following in the positive rational index form.

a. 5 b. 3 9 c. 4 243 d. 3

2

1 e. 6 x f. 3 3.yx g. 2yx

h. n px i. n qp yx . j.

3 2

4 2

y

x

k. 3 23 . yx


8. Evaluate.

a. 64 3

2

b. 125 3

2

c. 625 4

3

d. 81 4

3

e. 32

77

f. 3

1

125

1

g.

4

22

1

h.

5

4

32

1

i. 243 3

2

9. Simplify each of the following, giving your answer in the surd form.

a. 2

9

3 .

xx b. 4 33 2 . xx c.

3

1

3 2

y

x d.

ba

ba

ba.

.

. 2

5

3

e. 8 3 22 xxx f.

3 4 3 22 xxx g.

4

3

4

12

6

1

4

1

3

1

2

1

:.

x

y

x

yyx

h. 3

1

3 4 1234

1

aaaa i.

33

3 4

3

13

3

133.

1

13

xx

xxx

x

x

j.

3

113

22

3

1

3 23

2

3

2

3

11

.

cb

a

a

c

ca

ba k.

5

2

5

3

24

3

3

2

4

2

8

21627

l. 3522

2

1

2

1

2

1.

2

1.25,0.125,0.2.4

m.

33

11

36

1

2

114 9 823

2

..283.2..21.64 baba


10. Simplify each of the following surds.

a. 272523 b. 2

15018 c. 28

2

1

4

3172175

d. 4444 1623281

22.3 e. 3333

9

11923000

8

3.

5

1

f. 32

32

33 8.

2

127

a

b

b

aabab g. 3333

9

13000

8

3.5375

h. 55.4.22.3 i. 21.632 j. 7.65.27.55.3

k. 3333 64.32 l. baccba ... m. 75.3.75.3

n. 51.51 o. qpqpqp .. 4444

p. 2.. abba q. 25.26 r. 25.27.3

s. 357.2 t. 3.2..5 aba u. 22.33.26

v. 22 ........ babababaabba w. 333 25.45.1236.5..26

Rationalisation of the denominator:

The general form of conjugate surds are ba and ba . The product of a pair of conjugate surds is

always a rational number.

11. By rationalising the denominators, simplify:

a. 6

3607248 b.

3

211524 c.

3

43

3

222.5 d.

6

34

2

61218

e. 53

4

f.

3.24

3.2

g.

210

35.7

h.

35

35.2


i. 7.5.3

7.5.3

21

21

j.

yxxy

yxyx

..

..

k.

23

2.23.3

l.

33 ba

ba

m. 1

13

a

a n.

321

1

o.

73.25

35

p.

22322

21

q. 3 233 2 . bbaa

ba

r.

177

2

33 2 s.

333

33

469

23

baabba .2 and babaabba ;.2

12. Express in a b form,

a. 10.27 b. 15.28 c. 21.210 d. 78.219

e. 110.221 f. 130223 g. 2.46 h. 6.411

i. 5.614 j. 3.1452 k. 2.1027 l. 2.3055

m. 74 n. 32 o. 5.37 p. 70421

41

q. 65.327

13. a. 7474 b. 537537 c. 10.2910.29

d. 6.386.38 e. 53.5353.53 f. 33 13251325

14. Evaluate: 32.2

2

2.27

2.....

23

2

32

2

21

2

.


Advanced Exercise:

1. 12...12.12.12 512842

2. 13

1

13

1

13

1...

13

1

13

1

13

110009999989989991000

.

3. a. 481353.2 b. 4 13136497

4. a. 60402410 b. 7

23.8.3114

5. 5210.285210.28

6. 3222322232232

7. If x = 3819 , find 158

2318262

234

xx

xxxx.

8. 33

3

1.

3

8

3

1.

3

8

aaa

aaa .

9. Jika 123;123;123 zyx , maka

x2 + y

2 + z

2 + xy + yz + zx = ....

10. Jika x2 + 12x + 1 = 0, maka nilai dari

4

4 1

xx = ….

11. Rasionalkan penyebut:

a. 7523

5213515

b.

333

3

2427

116

12. Nilai x yang memenuhi 33

xxx adalah ….


13. Kurva xy 111 berpotongan dengan garis y = x di titik (a, b), maka nilai

a2 – b = ....

14. Nilai dari ...151413121 = ….

15. Bentuk sederhana dari:

a. 116116 b. 33 5252

LATIHAN BENTUK PANGKAT DAN AKAR

I. Jadikan bentuk √a + √b :

1. 6 2 5 2. 13 4 10 3. 10 2 21 4. 7 40

5. 6 2 4 2 3 6. 4 7 7. 6 2 5 8. 19 4 15

9. 12 2 35 10. 20 2 91 11. 7 4 3 12. 123 22 2

13. 80 28 10 14. 152 30 15 15. 7 3 5 16. 5 2 29 3 3

17. 4 57 24 3 18. 2 9 4 2

2 1

19. 32 10 7

20. 117 36 10 21. 28 5 12 22. 12

4 2 2

23. 2 3. 2 2 3 . 2 2 2 3 . 2 2 2 3 . . .


II. SEDERHANAKAN/HITUNGLAH :

1.)

514 1 6 232

16 5 3 3

( )a b a b c

a b c

. . .

2.)

2

22 15 10

5 3

. . .

3.) 12 2

1 1 1 1 22

2 2 2 2

1 2 21

:

a a a

aa aa a a a

. . .

4.)

11 1

1 1

x y xy

x y

. . .

5.)24 2 3

752 2 3

. . .

6.) 104 61

0,25 1,44 x10 22,5 10 243 1527

. . .

III. RASIONALKAN :

1.) 1

7 4 3

2.)

1

1 2 3

3.

33 3

7

16 12 9

IV. PILIHAN GANDA

1. Diketahui : 6x + y = 36 dan 6x + 5y = 216, maka harga x = . . .

a. 14

b. 34

c. 54

d. 32

e. ) 74

2. Jika xy = 7, maka nilai

2

2

( )

( )

2

2

x y

x y

. . .

a. 22 b. 27 c. 214 d). 228 e. 2196

3. Jika 3x – 3x – 3 = 78√3; maka nilai x = . . .

a. 3√3 b. 32

√3 c. 81√3 d). 92

e. 94


4. Jika 12( )x xa e e dan 1

2( )x xb e e maka nilai

22 2a b . . .

a. 2xe b. 2xe c. 2 2xe e d) 1 e. 0

5. 3 32 49 2169 3 8 12 64 8 50

13 16 5

a. – 29 b. – 11 c. 5 d. 17 e) 24

6. Nilai x yang memenuhi persamaan : 2 2

3 7

13

27

x

x

adalah :

a) 2,5 b. 2 c. 1 d. – 2,5 e. – 1,25

7. Nilai x yang memenuhi : 2 4

84 2

x

x adalah :

a. 15

2 b.

13

2 c.

11

2 d.

9

2 e.

7

2

“Saya tidak pernah meminta agar Tuhan menjadikan hidup ini mudah. Saya hanya meminta agar Ia

menjadikan saya kuat.”


LOGARITHMS

If a number (b) is expressed as the exponent c of a number (a), i.e. , cb=a a>0, a 1 , we say that c is

the logarithm of b to the base a. We write this as a logb=c , sometimes as alog b=c .

In general: c

b=a alogb = c , a>0, a≠0

For example, 2100log2100log10100 102 or 38

1log2

8

1 23

Exercise:

1. Convert the following to logarithm form:

a. 8134 b. 49

17 2 c. rpq

2. Convert the following to exponential form:

a. 532log2 b. 29log3 c. rqp log

3. Find the value of each of the following:

a. 64log2 b. 4log2

1

c. 1log3 d. 7log7

e. 25,0log8 f. )9log(3 g. 9log81 h. 32log22

4. Find x: a. 5

1164log x b.

5log5

15log2

x

x

= 1

Note: a. logarithms of a positive number may be negative

b. logarithms of 1 to any base is 0 i.e. 01log a

c. logarithms of a number to base of the same number is 1 i.e. 1log aa

d. logarithms of negative numbers are not defined, for example )4log(2

e. the base of a logarithm cannot be negative, 0 or 1. Can you think of why this is so ?

Laws of logarithms:

1. ba ba

log

2. n

mna

babm

log

3. cbcb aaa .logloglog

4. c

bcb aaa logloglog

5. bnb ana loglog

6. bb a

mnnam

loglog


5. Prove laws of logarithms no. 1 – 6.

6. Find the value of each the following:

a. 25log4

4 b. 2log5

5 c. 4log3

9 d. 6log125

25 e. 8log2

8

f. 5log27

3 g. 10log2

4 h. 7log625

55 i. 2

19 log381 j. 6log

41

8

2

7. Simplify and evaluate:

a. log 25 + log 4 b. 25log200log 44 c. 250log8log5log 5

2

5

2

5

2

d. 2 log 2 + 2 log 3 + 3

1 log 5 + log 7 – log 9 + log 10 + log 3 25.5 -

2

1 log 49

e. 144log.14log10log9log.7log5log 3

21333

3133

8. Expand to a single logarithm:

a.

2

3 .log

z

yx b.

53.log

z

yx c.

22

22

logyx

yx d. yxx .log 234

9. Given that log 2 = 0,3010 dan log 3 = 0,4771, find

a. log 0,002 b. log 3000 c. log 6000 d. log 15 e. log 3

4 f. log

24

5 g. log 3 6.3

10. Evaluate:

a. 2log

2log4log

3

3

9

2

627 b.

8log54log.516log2log.3

512log.8

c.

4,0log

5log2log 22

11. a. If my

xa

log , find 6 22 :log yxa .

b. If nxyb 3 :log , find 5 3:log yxb


Laws of logarithms:

7. a

bb

p

pa

log

loglog

8. ccb aba logloglog

9. naa bbn

loglog

10. a

bb

a

log

1log

12. Prove laws of logarithms no. 7 – 10.

13. Simplify:

a. 51352 log64log27log b.

43

2

loglog

log

bd

c

ac

b

a

14. Evaluate:

a. 81log

1

81log

118

2

1 b.

5log2

125log

25log

1

2

1

4

1

15. Simplify: 118log8log3log10log 252555

525

.

16. a. If p5log27 , find 55log243 .

b. Given p8log5 , find 125,0log2,0 .

c. Given a27log25 , find 5log9 .

d. If m27log16 , find 8log3 .

e. Given a5log4 , find 25,1log1,0 .

17. Given a3log4 , express the following in a:

a. 3log2 b. 81log8 c. 9116 log

18. Given ba pp 30log,5log and cp 21log . Express the following in a, b or c.

a. 211logp b. 10logp c. 36logp

19. Given m30log6 and n20log6 . Express 3log6 in m or n.


20. Simplify 9log9log

3log25

.

21. If a5log3 and b8log25 , find 750log15 .

22. For a, b and M are greater than 1, and xaM Mabb

loglog , find x.

23. Prove : bcabccba

abab

loglog .

24. Given

a

bc

a

cb

a b

b

c

c

ba log

log

log

log

log

11

2

. Prove a + b = c.

25. Given ayx 2log and by

xlog . Find xy log .

26. Given ba 7log,5log 22 and c5log3 . Express 98log48 in a, b or c.

27. Evaluate: 9log92

2

342

1

216log.3log5log10log

20log

15log

15log

.

28. Evaluate:

3 2 3 2 4 936

2 33 3

log 36 log 4 log125 log125. log5 5

log25. log25log12 . 144

.

INDICES AND SURDS (BILANGAN BERPANGKAT DAN BENTUK …€¦ · (BILANGAN BERPANGKAT DAN BENTUK AKAR)...

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