INDICES AND SURDS (BILANGAN BERPANGKAT DAN BENTUK ...

INDICES AND SURDS (BILANGAN BERPANGKAT DAN BENTUK AKAR)

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

a n  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,

54  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: 1

a o  1, a  p 

For example, 2 o  1, 2 3 

q

p 1 p , a p  a, a p  aq p a

1

3

1 1  , 5 4  4 5 and 7 5  5 7 3 23 8

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

a m x a n  a mn am  a m n an

a 

m n

 am n

    ruler for same base   

  a.b  a x b   ruler for same index n n  a a     b   bn    n

1 |matematika kls X (wajib)

n

n

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

2,

examples of irrational numbers are

3

7 ,  , etc. An irrational number involving a root is called a surd.

General rules involving surds:

p

n

a  q n

n

a 

a .

n

p  q

b 

n

n

. a

p

;

n

a.b ;

n

n

a  q

a  b

n

n

a 

p  q

n

a

a b

OVERVIEW LAWS OF INDICES a. b. c. d.

m

n

m+n

a xa =a am : an = am-n (am)n = amn a0 = 1 m

LAWS OF SURDS a.

x. x x

b.

x . y  xy

c.

a a x  x x

d.

x x  or y y

e. a n  a m n

1 f. a = n a n g. a x bn = (ab)n -n

a h. am : bm =   b i. a

m  n



a j.    b

n

a k.   b



m n

m

1 n

 b   a

e. a x  b x  (a  b) x f. a x  b x  (a  b) x g. h.

am n

i. m

 bn   a

j. k.


xy y

 x  y  x  y  x  y a x  b y  a x  b y   a x  b y a  x  y a  2

xy

x y

 

 y

x y

2

 x  2 xy  y

x

2

 x  2 xy  y

2

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

5

a. x 3

b. 6 4

1

f. x

2



2

.y

3



3

g. x

d.  x  y 

c. x  y  7

1 3



.y

1 4

h. x

2

1 5  3 .y 3





15

3

e. 7 2

5

15



4

i. x

.y

3 7

2. Evaluate the following without the use of calculator:

4  3

1

3

2

2

x2

a.

b.

d.

1

3 2 4  x5  

4  2

2

1

3

4

2

x9

c.

32

3



2

 1 5 3  

 1 5 2  

1

12   4

1

5 2 6  x5   

 2

3. If a  0 , a

f. 9

 3

1 2

4

3

e.   

x y

2n 1

x 31n : 27 n1

a yz a zx is equal ….

4. Simplify :

x 1  y 1 a.

b.

x 2  y 2

5. If 3 6.

3 n1  3 n 2

x 1

If 3  7

4 n 1  2 2n 3 c.

3 n1  3 n 2

4 n  2 2n 1

 2 , then 3  2x = …. x

and 7  3 y , find x . y

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

h.

5

n

xp

b.

3

9

i.

n

x p .y q


c.

4

243

4

x 2

3

2

j.

y

d.

k.

3

1 2

x 3 .3 y 2

e.

6

x

f.

3

x 3 .y

g.

x  y2

8. Evaluate.

a. 64

2 3

 1  f.    125 

b. 125



1 3



2 3

1  g.  2 2 

c. 625

4

3 4

d. 81



 1  h.    32 

4 5



i. 243

 

3 4

e. 7 7



2 3

2 3

9. Simplify each of the following, giving your answer in the surd form. 1

a.

e.

9  3 x .x 2

8

b.

3

x2

x2 x

f.

3

3

 3  x  c.   3 y 2   

x 2 .4 x 3

x 2

4

x

3

x2

g.

h. a

j.

1 4

3

1 1 a 3 2  a 3

 3 a 

b 3

c



3

a 2

2 3

3

.

2

c   a

2   a2  3

   b   

 4 1  a  

5

a.b 



2

1   y4  : 3  1  6 x4 x 

1

1 y4

3

.

3x 4

3x

1 x

3

1

 

 1 1 x 2 .y 3 

3x 

 a .3 b  2   a .b  d. 

i.

3x

3

1 x 3

1 x3

1 2

3

k. 27 3  16 4  1 1 c 3

2 2  8 3



5

4

2 

2 5

1

 1 2 l.  4 2  . 2 2    

 

2

.

5

0,125 .0,25.


1 31 . 2 2

m.

4

64.a



2 3

9

3

212.b 8 . 2

1

1 1  2 .3 6

1 1 3

28 3.a.b

10. Simplify each of the following surds. a. 3 2  5 2  7 2

d. 3.4 2  4

f.

2 4  32  4 162 81

a 1 8b  .3 b2 2 a2

ab  3 27ab  3

3



 2  3 . 4  6  3

3

3



3



n. 1  5 . 1  5



p. a. b  b. a



s. 2. 7  5

l.

 a  b. c . c.

e.

133 3 1 .  3000  3 192  3 5 8 9

g.

3





2

q. 6  2. 5





4











 3.

r.



3





m. 3. 5  7 . 3. 5  7

p 4 q . 4 p 4 q . p  q

2

t. 5. a.b  2. a

v. a. b  b. a . a 2 .b  a.b. a.b  a.b 2







3

3 3 1   3000  3 8 9

j. 3. 5  5. 7 2. 5  6. 7

o.



375  5.3 

a b









3 1 175  2 7  1  28 4 2

c.

i. 2 3  6 . 1  2

h. 3. 22 .4. 55

k.

1 2

18  50 

b.

u.



7  2. 5









2

 6  2. 3  3. 2 

2



w. 6  2..3 5 . 36  12.3 5  4.3 25



Rationalisation of the denominator: The general form of conjugate surds are

a  b and

a  b . The product of a pair of conjugate surds is

always a rational number.

11. By rationalising the denominators, simplify: a.

48  72  360

b.

24  15  21

6

e.

4 3 5


c.

3

f.

2. 3 4  2. 3

g.

5.3 2  2  4 2 3

7. 5  3 10  2

4

d.

18  3 12  6 6

3

h.

2. 5  3 5 3

2

i.

3. 5  12 . 7 3. 5 

m.

q.

1. 2

n.

a 1

1

r.

a 2  3 a.b  3 b 2

a  b  2.

3. 3  2. 2

35

3

3

s.

7 2  3 7 1

ab  a  b

3

a  b  2.

and

a 3 b

3

1  2

p.

5  2. 3  7

2

ab

l.

3 2

o.

1 2  3

ab 3

k.

y. x  x. y

7

a 1 3

x. y  x. y

j.

22 3 2 2

3 3 2

9 3 6 3 4

ab  a  b ; a  b

a  b form,

12. Express in a.

7  2. 10

b.

8  2. 15

c.

10  2. 21

d.

19 2. 78

e.

21 2. 110

f.

23 2 130

g.

6  4. 2

h.

11 4. 6

14 6. 5

j.

52 14. 3

k.

27 10. 2

l.

55  30. 2

m.

4 7

n.

2 3

o.

7  3. 5

p.

4 14  12 70

q.

27 3. 65

i.

13. a. d.

4 7  4 7

b.

8  3. 6  8  3. 6

e. 3  5 . 3  5  3  5 . 3  5

14. Evaluate:

2 1 2



2 2 3




7 3 5  7 3 5





2

 .....

32



2 7  2. 2





9  2. 10  9  2. 10

c.

2 2. 2  3

f.

.

3

5  2 13  3 5  2 13

Advanced Exercise: 1.

2.

2

2



1 3







 1 . 24  1 . 28  1 ... 2512  1 

1000

1 3

1 999

1



1 3

998

1

3. a. 2. 3  5  13  48

b.

10  24  40  60

4. a.

1

 ... 

b.

998

3

11  3. 8 . 3  2 7

2 3

8.

3

a

9. Jika 2

2 2 2 3

1

1 3

4

6.

19  8 3 , find



497136 13

8  2. 10  2 5  8  2. 10  2 5

7. If x =

1 3

999

4

5.

2 2 3

1



1000

1

.

2 2 2 3

x 4  6 x 3  2 x 2  18x  23 . x 2  8x  15

a  8 a 1 3 a  8 a 1 .  a . 3 3 3 3

.

x  3  2  1 ; y  3  2  1 ; z   3  2  1 , maka 2

2

x + y + z + xy + yz + zx = .... 2

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

11.

1 x4

= ….

Rasionalkan penyebut:

a.

12.

x4 

15  35  21  5 32 5 7

Nilai x yang memenuhi


x

xx

b.

3

 3 adalah ….

3 16  1 3 27  3 4  3 2

13.

Kurva

y  1  1  1  x berpotongan dengan garis y = x di titik (a, b), maka nilai

2

a – b = ....

1  2 1  3 1  4 1  5 1  ...

14.

Nilai dari

15.

Bentuk sederhana dari:

a.

= ….

6  11  6  11

b. 3

2 5 3 2 5

LATIHAN BENTUK PANGKAT DAN AKAR I. Jadikan bentuk √a + √b : 1. 5.

62 5

62 42 3

9. 12  2 35 13.

80  28 10

17.

4  57  24 3

20. 117  36 10 23.

2. 13  4 10 6. 10.

3. 10  2 21 7.

20  2 91

11.

74 3

12. 123  22 2

15.

7 3 5

16.

18.

62 5

2 94 2

28  5 12

19.

2 1 22.

8. 19  4 15

5 9

 32

32 10 7

4 12  2 2

2  3. 2  2  3 . 2  2  2  3 . 2  2  2  3  . . .


7  40

4 7

14. 152  30 15

21.

4.

2 3

II. SEDERHANAKAN/HITUNGLAH : 5

1

1.)

(a 4b1 ) 2 a6b 3 c 2

a b



1 6 5 3 3

c

 . . .

2

 2  2.) 2 15     10  . . .  5 3

 2 a2  a  1  a2 3.)  1   1 1 1 2  a 2  a 2 a : a a 2  a 2 

 1  2  1   . . .  2  a    

1

 x1 y  xy 1  . . . 4.)  1 1   x y  5.)

24 2  3   75  . . . 2 2 3

6.) 0, 25 4 1, 44 x10 22,5 1010 243 15 6

1 ... 27

III. RASIONALKAN : 1.)

1 74 3



2.)

1  1 2  3

3.

3

7  16  12  3 9 3

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

1 4

b.

3 4

c.

5 4

d.

3 2

e. )

7 4

2( x  y ) 2. Jika xy = 7, maka nilai ( x  y )2  . . . 2 2

a. 22

b. 27

c. 214

d). 228

e. 2196

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

b.

3 2

√3


c. 81√3

d).

9 2

e.

9 4

4. Jika a  12 (ex  e x ) dan b  12 (ex  e x ) maka nilai a. e2x

5.

a

c. e2 x  e2

b. e2x

2

 b2   . . . 2

d) 1

e. 0

d. 17

e) 24

2 49 3 2 169  3 3 8 12  64  8  50  13 16 5 a. – 29

b. – 11

c. 5

6. Nilai x yang memenuhi persamaan : a) 2,5

b. 2

7. Nilai x yang memenuhi : a. 

15 2

b. 

13 2

1  32 2 x adalah : 3x 7 27

c. 1

d. – 2,5

2 4x

4x  8 adalah : 2

c. 

11 2

d. 

9 2

e. – 1,25

e. 

7 2

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


LOGARITHMS b = a c  a >0, a  1 , we say that c is

If a number (b) is expressed as the exponent c of a number (a), i.e. , a

the logarithm of b to the base a. We write this as

b = ac

In general:

For example, 100  10 2 

a

 10

logb=c , sometimes as loga b=c .

logb = c

a > 0, a ≠ 0

,

1  2 3  8

log 100  2 or log 100  2

2

log

1  3 8

Exercise: 1. Convert the following to logarithm form: 1 a. 34  81 b. 7 2  c. p q  r 49 2. Convert the following to exponential form: a. 2 log 32  5 b. 3 log 9  2 c. p log q  r 3. Find the value of each of the following: 1 2

a. 2 log 64

b.

e. 8 log 0,25

f. 3 log(9)

4. Find x: a.

x

log 64  1

c. 3 log 1

log 4

g.

1 5

81

2

b.

d. 7 log 7 h.

log 9

x

log 5  1

2 2

log 32

=1

5  log 5 Note: a. logarithms of a positive number may be negative x

a

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

log 1  0

c. logarithms of a number to base of the same number is 1 i.e. d. logarithms of negative numbers are not defined, for example

a

2

log a  1

log(4)

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

a

log b

 

2. a m

 b m

log b

 bn

3.

a

log b 

a

log c 

a

log b.c

4.

a

log b 

a

log c 

a

log

5.

a

log b n  n

6.


an

am

log b n 

a

log b

n a log b m

b c

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

6. Find the value of each the following: a. 4 f.

4

log 25

 3

27

5

b. 5

log 5

g. 4

log 2

2

c. 9

3

log 4

 

log 10

h. 5 5

625

d. 25

125

log 6

81 3 

9

log 7

i.

e. 8 log 12

j.

2

log 8

 2 1 4

8

log 6

7. Simplify and evaluate: a. log 25 + log 4

3

log 200 

4

log 25

c.

2 5

log 5 

2 5

log 8 

2 5

log 250

1 1 log 5 + log 7 – log 9 + log 10 + log 5.3 25 - log 49 2 3

d. 2 log 2 + 2 log 3 +

e. 3 log 5 

4

b.

log 7 

1 .3 log 9 3

 3 log 10  3 log 14 

1 .3 log 144 2

8. Expand to a single logarithm:

 x. y    z2   

  

a. 3 log

b. log x 3 .5

y  z 

 x2  y2  x2  y2 

c. log

   





5 24

g. log 3.3 6

d. 4 log x 3  x 2 . y

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

4 3

f. log

10. Evaluate: a. 27

9

log 4



6 2

11. a. If

a

b. If

b

3 3

log 2 log 2

x log   m , find  y log

3

a

log

y : x  n , find


8. log 512 3. log 2  log 16  5. log 4  5 log 8

b.

b

6

log

x2 : y2 . 5

x : y 3

c.

log 2 2  log 2 5 log 0,4

Laws of logarithms: a

7.

log b 

a

log b

9.

a

log b  a

log b log a

log c 

8.

10.

b

p p

an

log b 

a

log c

log b n

b

1 log a

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

13. Simplify: 2

a.

log 27

5

3

log 64 log



 b 2 b.  a log c 

1 5



c

log d 3

  

a

log b4

14. Evaluate: a.

1 1 2



log 81



18

1 log 81

16. a. If

27



5

15. Simplify: 25 5

b.

log105 log 3 25log 8 25log181

log 5  p , find

243

b. Given log 8  p , find 25

d. If

16

4

log 25

1 2 2 log 5

.

0, 2

log 0,125 .

log 27  a , find 9 log 5 .

log 27  m , find

e. Given

 log 1 25 

log 5 5 .

5

c. Given

1 1 4

3

log 5  a , find

log 8 . 0,1

log 1,25 .

17. Given 4 log 3  a , express the following in a: b. 8 log 81

a. 2 log 3

18. Given a.

p

p

log 5  a,

log 1 12

p

log 30  b and b.

p

log 10

c.

p

16

log 19

log 12  c . Express the following in a, b or c. c.

p

log 36

19. Given 6 log 30  m and 6 log 20  n . Express 6 log 3 in m or n.


20. Simplify

2 log 3 5

.

log 9  log 9

21. If 3 log 5  a and

25

log 8  b , find

15

log 750 .

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

23. Prove : c

24. Given 1 

ab

log a

abc

ab

log b

b

log M

b

log a

 M x , find x.

 bc .

c logb  c  b logc  b 1 . Prove a + b = c.   b a  2b c log a log a log a

 

25. Given log x 2 y  a and log 26. Given 2 log 5  a,

2

x  b . Find y

y

log x .

log 7  b and 3 log 5  c . Express

48

log 98 in a, b or c.

1

4 log 3 5 1 2 log 20 27. Evaluate:  2  log 2 5  log 3.9 log 16  2 log 9 . log 5 1 log 10

3

28. Evaluate:

 4 log125  9 log125  36 log2 36  3 log2 4   2 log25. 3 log25  . log5 5 . 3 log12 . 3 144  
