INDICES AND SURDS (BILANGAN BERPANGKAT DAN BENTUK AKAR)

1 |matematika kls X (wajib)

INDICES AND SURDS

(BILANGAN BERPANGKAT DAN BENTUK AKAR)

Where n is a positive integer, an 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,

5

4



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 p q

q p p p p o

a a a a a a

a 1,   1 ,  , 

1

For example, 5 5 3

3 4

4 1 3

3 _{, 5 5 7 7}

8 1 2

1 2 , 1

2o      and 

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

m m n n

n

m m n

a x a

a

a

a

ruler for same base

a

a

a

























 

n _{n n}

n _n

n

a.b

a x b

ruler for same index

a

a

b

b













  



 

2 |matematika kls X (wajib)

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

examples of irrational numbers are 2, 37,



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

General rules involving surds:





n n n

p

a



q

a



p q



. a

;

p

n

a



q

n

a





p q





n

a

n

a .

n

b



n

a.b

; n

n n

a

a

b

b



OVERVIEW

LAWS OF INDICES

LAWS OF SURDS

a. a

m

x a

n

= a

m+n

b. a

m

: a

n

= a

m-n

c. (a

m

)

n

= a

mn

d. a

0

= 1

e.

m

n m n

a



a

f. a

-n

=

1

_n

a

g. a

n

x b

n

= (ab)

n

h. a

m

: b

m

=

m

a

b

 

 

 

i.

m n

n m

1

a

a





j.

n n

a

b

b

a



 



 

 

 

 

 

k.

m m

n n

a

b

b

a



 



 

 

 

 

 

a.

x . x



x

b.

x . y



xy

c.

a

a x

x

x



d.

x

x

or

xy

y

y

y



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 2

a x



b y

a x



b y



a x b y



i.

a

a



x

y



x

y

x

y









3 |matematika kls X (wajib)

Exercise:

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

a. 3 1

x b. 4 5

6 c.





7 3 y

x d.





3 5

x y



 e. 2 15 7

f.

2 1

3 2

x

. y

 g.

1 1

3 4

x

. y

 

h. 3 5 3 1 2

.y

x i.

3 15

7 4

x

. y

 

2. Evaluate the following without the use of calculator:

a.

 

3 1

3 ₂ 2

3 2

4

x 2

2



b.

1 1 ₂ 3

2 4

2

5

x 5

3



















c.

 

3 1

2 _{4 2}

1 2

4

x 9

1

4

 

 

 

d.

1

1 ₂ 5

3 6

1 2

5

x 5

4



















e.

3

2

3

















f.

2n 1 1 n n 1

9



x 3



: 27



3. If

a 0 , a



x y

a

y z

a

z x 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

3

x 1



2

, then

3

2x= …. 6. If

3 7



x

and 7



3

y, find x . y

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

a. 5 b. 39 c. 4243 d. 3 2 1

e. 6x f. 3 _x3_{.y g. xy}2

h. nxp i. n_xp_.yq _j.

3 2 4 2

y x

4 |matematika kls X (wajib)

8. Evaluate.

a. 643 2

b. 125 3 2 

c. 6254 3

d. 81 4 3 

e.

 

3 2 7 7

f. 3 1 125 1        g. 4 2 2 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_.x

x b. 3 x2.4x3 c.

3 1

3 2         y x

d.               b a b a b a . . . 2 5 3

e. 8 x2 3 x2 x f. 3 x2 4x 3x2 g.

4 3 4 1 2 6 1 4 1 3 1 2 1 : . x y x y y x          

h. 3

1 3 2 4 1 3 4 1              a a a

a i.

3 3 3 4 3 1 3 3 1 3 3 . 1 1 3 x x x x x x x j. 3 1 1 3 2 2 3 1 3 2 3 2 3 2 3 1 1 .                     c b a a c c a b a k. 5 2 5 3 2 4 3 3 2 4 2 8 2 16 27     

l. 2

 

2 2 5 3 1 2 1 2 1 . 2 1 . 25 , 0 . 125 , 0 . 2 . 4 _  

     

m. 3 3

1 1 3 6 1 2 1 1

4 _{9 2 8}

3 2 . . 28 3 . 2 . . 21 .

5 |matematika kls X (wajib)

10. Simplify each of the following surds.

a. 3 25 27 2 b.

2 1 50

18  c. 28 2 1 4 3 1 7 2

175  

d. 4 ₄ 4₃₂ 4₁₆₂ 81

2 2 .

3    e. 3 3 3 3 9 1 192 3000 8

3 . 5

1 _{  }

f. 3

2 3

2 3

3 _. 8

2 1 27

a b b

a ab

ab   g. 3 3 3 3 9 1 3000 8

3 . 5

375    

h. 3. 22.4. 55 i.



2 3 6

 

.1 2 j.



3. 55. 7



2. 56. 7



k.



3233



.3436



l.



ab. c



.c. a b



m.



3. 5 7



.3. 5 7



n.



1 5

 

.1 5 o.



4 p4q



.4 p4q



. p q



p.



a. bb. a



2 q.



62. 5



2 r.



3. 72. 5



2

s.



2. 7 5



3 t.



5. a.b2. a



3 u.



62. 33. 2



2

v.



a. bb. a



.a2.ba.b. a.ba.b2



w.



_62..3₅



_.3612.3_54.3₂₅



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.

6 360 72

48 

b.

3 21 15

24 

c. 3

4 3

3 2 2 2 .

5  

d. 6 3 4

2 6 12

18 

e. 5 3

4

 f. 4 2. 3 3 . 2

 g. 10 2 3 5 . 7

 

h.

3 5

3 5 . 2

6 |matematika kls X (wajib)

i.

7 . 5 . 3

7 . 5 . 3

2 1 2 1

 

j.

y x x y

y x y x

. .

. .

 

k.

2 3

2 . 2 3 . 3

 

l. 3 3_{a b}

b a

 

m. 1 1 3 _  a a

n.

3 2 1

1 

 o. 5 2. 3 7 35



 p. 2 2 3 2 2 2 1

 

 

q.

3 2 3

3 2

.b b a a

b a

 



r.

1 7 7

2 3

3 2 _{ } s. 3 3 3 3 3

4 6 9

2 3

 





a



b





2

.

ab



a



b

and



a



b





2

.

ab



a



b

;

a



b

12. Express in

a



b

form,

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

i. 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 ₂1 70

4

1

q. 273. 65

13. a.

4



7



4



7

b.

7



3

5



7



3

5

c. 92. 10 92. 10

d. 83. 6 83. 6 e.



3 5



. 3 5



3 5



. 3 5 f. 352 13352 13

14. Evaluate:

3 2 . 2

2 2 . 2 7

2 ... 2 3

2 3 2

2 2 1

2

        

7 |matematika kls X (wajib)

Advanced Exercise:

1.



2

2



1



.

2

4



1



.

2

8



1

 

...

2

512



1



2.

1

3

1

1

3

1

1

3

1

...

1

3

1

1

3

1

1

3

1

1000 999

998 998

999

1000





























 .

3. a. 2. 3 5 13 48 b. 4497136 13

4. a. 10 24 40 60 b.

7

2

3

.

8

.

3

11

4





5.

8



2

.

10



2

5



8



2

.

10



2

5

6. 2 3 2 2 3 2 2 2 3 2 2 2 3

7. If x =

19



8

3

, find

15

8

23

18

2

6

2 2 3 4













x

x

x

x

x

x

.

8. 3 3

3

1

.

3

8

3

1

.

3

8















a

a

a

a

a

a

.

9. Jika

x



3



2



1

;

y



3



2



1

;

z





3



2



1

, maka

x2 + y2 + z2 + xy + yz + zx = ....

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

4 4

1

x

x



= ….

11. Rasionalkan penyebut:

a.

7

5

2

3

5

21

35

15











b.

3 3 3

3

2

4

27

1

16







12. Nilai x yang memenuhi

3

3



x

x

8 |matematika kls X (wajib)

13. Kurva

y



1



1



1



x

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

a2– b = ....

14. Nilai dari

1



2

1



3

1



4

1



5

1



...

= ….

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.

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 2

9



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.

1

2

4



2 2

9 |matematika kls X (wajib)

II. SEDERHANAKAN/HITUNGLAH :

1.)





5 1

4 1 ₂ 6 ₃ 2 1 6 5 _{3 3}

(

a b

)

a b c

a b c

 

 



. . .

2.)

2

2

2 15

10

5

3





















. . .

3.)

1

2 2

1 1 2 1 1 2

2 2 2 2

1

2

2

1

:

a

a

a

a

a

a

a

a

a

a



 

 













































. . .

4.)

1

1 1

1 1

x y

xy

x

y



 

 



















. . .

5.)

24

2

3

75

2

2

3











. . .

6.)

0, 25 1, 44 x10 22,5 10 243 15

4 10 6

1

27







. . .

III. RASIONALKAN :

1.)

1

7 4 3





2.)

1

1



2



3



3.

3 3 3

7

16



12



9



IV. PILIHAN GANDA

1. Diketahui : 6

x + y

= 36 dan 6

x + 5y

= 216, maka harga x = . . .

a.

1

4

b.

3

4

c.

5

4

d.

3

2

e. )

7 4

2. Jika xy = 7, maka nilai

2

2

( ) ( )

2

2

x y x y







. . .

a. 2

2

b. 2

7

c. 2

14

d). 2

28

e. 2

196

3. Jika 3

x

–

3

x – 3

= 78√3; maka nilai x = . . .

a. 3√3

b.

3

2

√3

c. 81√3

d).

9

2

e.

10 |matematika kls X (wajib)

4. Jika

1

2

(

)

x x

a



e



e



dan

1

2

(

)

x x

b



e



e



maka nilai



a

2



b

2



2



. . .

a.

e

2x

b.

e

2x

c.

e

2x



e

2

d) 1

e. 0

5.

2

169 3

3

8 12

49

3

64

8

2

50

13

  

16

  



5



a.

–

29

b.

–

11

c. 5

d. 17

e) 24

6. Nilai x yang memenuhi persamaan :

1

_{3 7}

3

2 2

27

x x







adalah :

a) 2,5

b. 2

c. 1

d.

–

2,5

e.

–

1,25

7. Nilai x yang memenuhi :

2

4

8

4

2

x

x



adalah :

a.

15

2



b.

13

2



c.

11

2



d.

9

2



e.

7

2



“

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11 |matematika kls X (wajib)

LOGARITHMS

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

b = a

c



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

log b=c

_a .

In general:

b = a

c  a

logb = c

,

a > 0, a

≠ 0

For example, 100102  10log1002 or log1002 3 8 1 log 2

8

1_ 3 _ 2 _

Exercise:

1. Convert the following to logarithm form: a. 34 81 b.

49 1

72  c. pqr

2. Convert the following to exponential form:

a. 2log325 b. 3log92 c. plogqr 3. Find the value of each of the following:

a. 2log64 b. 2log4 1

c. 3log1 d. 7log7 e. 8log0,25 f. 3log(9) g. 81log9 h. 2 2log32

4. Find x: a.

5 1 1 64

log 

x _b.

5 log 5

1 5 log 2

x x

 

= 1

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

b. logarithms of 1 to any base is 0 i.e. alog10

c. logarithms of a number to base of the same number is 1 i.e. aloga1 d. logarithms of negative numbers are not defined, for example 2log(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. aalogb  b

2.

 

n

m n

a

b am logb 

3. alogb  alogc  alogb.c 4.

c b c

b a a

a

log log

log  

12 |matematika kls X (wajib)

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

6. Find the value of each the following:

a. 44log25 b. 55log 2 c. 93log4 d. 25125log6 e. 82log8

f.

 

log5

27

3 g. 4 2log10 h.

 

log7

625 5

5 i.

 

2 1 9

log

3

81 j.

 

₄1 log6

8 2

7. Simplify and evaluate:

a. log 25 + log 4 b. 4log200  4log25 c. log5 log8 5log250 2 5

2 5

2

 

d. 2 log 2 + 2 log 3 + 3 1

log 5 + log 7 – log 9 + log 10 + log 5.325 -2 1

log 49

e. log5 log7 .3log9 3log10 3log14 ₂1.3log144 3

1 3

3 _{    }

8. Expand to a single logarithm:

a.

       

2

3 .

log z

y x

b. _   

  

5 3

. log

z y

x c. _   

  

 

2 2

2 2

log y x

y x

d. 4log



x3x2.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 3 4

f. log 24

5

g. log_3.3₆

10. Evaluate:

a.

2 log

2 log 4 log

3 3 9

2 6

27  b.

8 log 5 4 log . 5 16 log 2 log . 3

512 log . 8

 

 c. log 0,4 5 log 2 log2  2

11. a. If m y x

a _

     

log , find 6 2 2 :

log x y

a _.

b. If b_log 3 y_:xn _{, find} 5

 

3 :

log x y

13 |matematika kls X (wajib)

Laws of logarithms:

7.

a b b

p p a

log log

log 

8. alogb blogc  alogc

9. alogb  anlogbn

10.

a b

b a

log 1

log 

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

13. Simplify:

a.

5 1 3 5

2_{log27 log64 log b.}

 

4 3 2

log log log

b d c

a c b a

   

 

14. Evaluate: a.

81 log

1 81 log

1

18

2

1  b.

5 log 2

1 25 log 25 log

1

2 1

4

1  



15. Simplify:

 

log10 log3 log8 log181

25 25 5 5 5

25     .

16. a. If 27log5p, find 243log5 5. b. Given 5log8p, find 0,2log0,125. c. Given 25log27a, find 9log5. d. If 16log27m, find 3log8. e. Given 4log5a, find 0,1log1,25.

17. Given 4log3a, express the following in a: a. 2log 3 b. 8log81 c. 16 ₉1

log

18. Given plog5a, plog30b and p_log1₂c_{. Express the following in a, b or c.} a. p_log121 _b. p_{log10 c.} p_log36

14 |matematika kls X (wajib)

20. Simplify

9 log 9 log

3 log 2

5 _ .

21. If 3log5a and 25log8b, find 15log750.

22. For a, b and M are greater than 1, and ablogM bloga Mx, find x.

23. Prove : cabloga

 

abc ablogb bc.

24. Given









a b c a

c b

a b

b c

c b

a _log

log log

log log

1 1

2

 

 

 _ . Prove a + b = c.

25. Given log

 

x2y a and b y x _

log . Find ylogx.

26. Given 2log5a, 2log7b and 3log5c. Express 48log98 in a, b or c.

27. Evaluate: 2 9 log9

2

3 2 ₄

1

2 16 log . 3 log 5 log 10 log

20 log 1 5 log

1 5

log _{   }

 

.

28. Evaluate:

3 2 3 2 4 9

36

2 3

3 3

log 36

log 4

log125

log125

.

log5 5

log25 . log25

log12 . 144









 

