INDICES AND SURDS (BILANGAN BERPANGKAT DAN BENTUK AKAR)

  

INDICES AND SURDS

(BILANGAN BERPANGKAT DAN BENTUK AKAR)

n a

  Where n is a positive integer, is defined as: _n a  a x a x a x ..... x a n faktor where a is called the base, and n, the index or exponent or power. ₄ For example, 5  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 q

  1 p

  p o  p p p q a

  1 , a , a a , a a    

  p a

  1

  

3

  1

  1

  

  3

  5

  3 o

  4

  4

  

5

  2  1 , 2   , 5  5 and 7 

  7 For example,

  3

  8

  2 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 } 

  a x a a _m   a _{m n } a

   _n  ruler for same base a  _{m m n n}   a  a

     _{n n n}   a.b  a x b

     ruler for same index _{n n}   a a   

    n  b b    A number that cannot be expressed as a fraction of two integers is called an irrational number. Some

  3 examples of irrational numbers are

  2 , 7 ,  , etc. An irrational number involving a root is called a surd.

  General rules involving surds: _{n n n n n n}

p a q a p q . a p a q a p q a

   ;   

_{n n n a a}     _{n n}

   ;  a . b a.b _n b b

  

OVERVIEW

LAWS OF INDICES LAWS OF SURDS

m n m+n

  a. a x a = a x . x  x a. m n m-n

  b. a : a = a

  b. x . y  xy m n mn

  c. (a ) = a a a x

  d. a = 1 _m c.  x _{n n m} x

  e. a  a x x xy or d. 

  1

• n

  y y y f. a = _n a n n n

   b x   (a b) x

  e. a x

  g. a x b = (ab) _m  b x   (a b) x f. a x a m m  

  h. a : b =   b   x  y x  y   x y g. _m

      _n

  1 _{2 2} i. a  _n a x b y a x b y a x b y _m

  h.        a

   n n a x y  a b a    

    j.  i. 

      b a x y x  y     _{m m}  ₂

   _{n n} a b x  y   x 2 xy  y     j.

     k.     b a     ₂ k. x  y   x 2 xy  y

   

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

  2 

  3

  5 b.

  a.

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

  3   , find x . y

  6. If ^{x y} 3 7 and 7

  3  = ….

   , then ^2x

  3

  4

  5. If ^{x 1}

   

  2   

  4

  2

  4

    c. ^{n 1 2n 3} _{n 2n 1}

  3    

  9 c.

  243 d.

  3

  n p x i. n q p y x . j.

  3 .

  2

  3

   k.

  2 y x

  4

  2

  3

   h.

  

3

  2 y x

  x g.

  .y

  3

  3

  6 x f.

  1 e.

  2

  3

  3

  a.

  2

  1

  3

  5

  3

    h.

  x . y  g. ^{1 1 3 4} x . y

  7 f. ^{2 1 3 2}

  15

    e.

   y x i. ^{3 15 7 4} x . y

    ^{3 5} x y

   d.

  c.   ^{7 3} y x

  6

  5

  4

  1 x b.

  3

  2 .

   

        b. ^{n 1 n 2} _{n 1 n 2}

  4 

  4. Simplify : a. ^{1 1} _{2 2} x y x y

      is equal ….

  3. If ^{x y y z z x} a 0 , a a a

  9 x 3 : 27   

         f. ^{2n 1 1 n n 1}

  3 

  2

          e. ³

  5 x 5

  2. Evaluate the following without the use of calculator: a.

  4       d. ^{1 1 5 2 3 6} _{1 2}

  1

    ^{3 1 2 2 4} _{1 2} 4 x 9

          c.

  3 

  5 x 5

  2  b. ^{1 1 3 2 2 4} ₂

    ^{3 1 3 2 2} _{3 2} 4 x 2

   y x

  8. Evaluate.

  1

  3

  1

  3

  2

  3

  2

  3

  2

  3

  1 .

  2

     

   

     

     

    

    

  c b a a c c a b a k.

  5

  2

  2

  3

  3

  3

  4

  1     

    

      

  a a a a i.

  3

  3

  3

  4

  1

  1

  3

  3

  1

  3 3 .

  1

  1

  3

  x x x x x x x j.

  3

  1

  5

  2

  2

  1

   

     

     

m.

  3

  3

  1

  1

  3

  6

  2

  1 . . 25 , 125 , . 2 .

  1

  1

  4

  9

  8

  2

  3

  2 . .

  28 3 . . 2 . 21 . 64 b a b a

  4

  2

  4

   

  3

  3

  2

  4

  2

  8

  2

  16

  27

     l.

  1 .

   

  3

  5

  2

  2

  2

  1

  2

  1

  2

  3

  1

  a. 64

    

  1

  

    

     h.

  5

  4

  32

  1

  

    

  2

  i. 243

  3

  2 

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

  a.

  2

  9

  3 .

   x x b.

  2

  4

  3

  3  e.

  3

  2

  b. 125

  3

  2  c. 625

  4

  3

  d. 81

  4

   

     g.

  3

  2

  7

  7 f.

  3

  1

  125

  1

  

    

  4

  3

  4

  1

  2

  2 x x x

   g.

  4

  3

  4

  1

  2

  6

  4

  4

  1

  3

  1

  2

  1 : . x y x y y x

      

       h.

  3

  1

  3

  

3

  3

  2

    

  . x

  x c.

  3

  1

  3

  2

     

     

  y x d.

    

  2 x x x f.

     

     

  b a b a b a .

  . .

  2

  5

  3 e.

  8

  3

  2

      

  10. Simplify each of the following surds.

  1

  3

  1 a.

  3 2 

  5 2 

  7

  c. 175 

  2 7  1 

  28

  2 b. 18  50 

  2

  4

  2

  2

  1

  3

  1

  4

  4

  4

  3

  3

  3

  3

  2 32 162 . 3000 192

  d.   

  4 3 .

  e.   

  81

  5

  8

  9

  a

  1 8 b

  3

  1

  3

  3

  3

  3

  3

  3

  3

  3 ab 

  27 ab   . 375  5 .    3000  f.

  g.

  2

  2

  2

  8

  9

  b a h.

  3 . 22 . 4 . 55 i.

  2 3  6 . 1  2 j. 3 . 5  5 .

  7 2 . 5  6 .

  7

       

  3

  3

  3

  3

    a  b c c a  b   k.

  2 3 .

  4 6 l. . . . m. 3 .

  5 7 . 3 .

  5

  7

          

  4

  4

  4

  4 p q . p q . p q n. 

  1  5 . 1  5 o.   

        

  2

  2

  2 p. a b  b a q.  r.   . .

  6 2 .

  5 3 .

  7 2 .

  5

       

  3

  

3

  2 a b a s.

  2 . 7  5 t. 5 . .  2 . u. 6  2 . 3  3 .

  2

       

  2

  2

  3

  3

  3 a b  b a a b  a b a b  a b    v. . . . . . . . . w.

  6 2 .. 5 .

  36 12 .

  5 4 .

  25

        Rationalisation of the denominator:

a  b a  b

  

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

always a rational number.

  11. By rationalising the denominators, simplify:

  3

  4

  4

  3

  48 72 360

  24

  15

  21 5 .

  2

  2

  2

  18

  12

  6         a.

  b.

  c.

  d.

  3

  6

  6

  3

  3

  2

  4 2 .

  3 7 .

  5

  3 2 .

  5

  3   e.

  f.

  g.

  h.

  3

  5

  4 2 .

  3

  10

  2

  5

  3    

  1 3 .

  5  . 7 x y x y .  . 3 .

  3 2 . 2 a b  

  2 i. j. k. l.

  3

  3

  1 y x x y

  3 2 a b 3 . 5  . 7 .  .  

  2 a 

  1

  1

  35

  1

  2   m. n. o. p.

  3 a 

  1 1  2 

  3 5  2 . 3 

  7 2 

  2 3 

  2

  2

  3

  3 a b

  2

  3

  2 

   q. r. s.

  3

  3

  3

  3

  2

  3

  3

  2

  3

  2

  

3

  9

  6

  4  

  a a . b b

  7

  7

  1    

  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. 

  f.  g.

  6  4 .

  2

  h. 

  21 2 . 110

  23 2 130

  11 4 .

  6

  27 10 .

  2

  55 30 .

  2 i. 14  6 . 5 j. 52  14 . 3 k.  l. 

  1

  1 m.  n.  o.  p. 

  4

  7

  2

  3

  7 3 .

  5

  4

  70

  4

  2 q.

  27  3 .

  65 13. a.

  4  7  4 

  7 b. 7 

  3

5 

7 

  3

  5

  c.   

  9 2 .

  10

  9 2 .

  10

  3

  3 d.

  8  3 . 6  8  3 .

  6 e. 3  5 . 3  5  3  5 . 3 

  5 f. 5 

  2 13  5 

  2

  13

     

  2

  2

  2

  2

  2 .....

  14. Evaluate:      .

  1

  2

  2

  3

  3

  2

  7 2 .

  2 2 .

  2

  3     

  Advanced Exercise:

  2

  4 8 512 1. 2  1 . 2  1 . 2  1 ... 2 

  

1

     

  1

  1

  1

  1

  1

  1    ...   

  2.

  .

  

 1000  999  998 998 999 1000

3 

  1 3 

  1 3 

  

1

3 

  1 3 

  1 3 

  1

  4 2 .

  3

  5

  13

  48

  3. a.   

b. 497  136

  13

  4 11  3 . 8 . 3 

  2 4. a.

  10  24  40 

  60 b.

  7 5. 8  2 . 10 

  2 5  8  2 .

10 

  2

  5

  2 

  3 2  2 

  3 2  2  2 

  3 2  2  2 

  3 6.

  4

  

3

  2 x 

6 x 

2 x  18 x 

  23

  8 3 , find .

  7. If x = 19 

  

2

x 

8 x 

  15 a  8 a  1 a  8 a 

  1

  3

  8. .

  3 a  .  a  .

  3

  3

  3

  3

  9. Jika x  _{2 2 2} 3  2  1 ; y 

3 

2  1 ; z   3  2  1 , maka x + y + z + xy + yz + zx = .... ₂

  1

  4 x 

  4 x

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

  11. Rasionalkan penyebut:

  3

  15  35  21 

  5 16 

  1 a. b.

  3

  3

  3

  3 

  2 5 

  7 27  4 

  2

  3 x x x 

  3

  12. Nilai x yang memenuhi adalah ….

  13. Kurva y  ₂ 1  1  1  x berpotongan dengan garis y = x di titik (a, b), maka nilai a – b = ....

  2 1 

  3 1 

  4 1 

  5 1  ...

  14. Nilai dari 1 

  = ….

  15. Bentuk sederhana dari:

  3

  3 a.

  6  11  6 

  11 b. 2  5  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 

  5

  2

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

  

  9

  3

  3 2  9 4 2  17. 4  57 24 3  18. 19. 32 10 7  2 

  

1

₁ 20. 117 36 10  21. 28 5 12  22. 4  2 2 ₂ 23. 2  3. 2 

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

II. SEDERHANAKAN/HITUNGLAH :

  1

  5 4  1 6 

  2

  3

  2 ( a b ) a b c 1.)  . . .

  1 6  5  3 3 a b c

   

  2  

  2 2.) 2 15   10  . . .   5 

  3  

  

  2 1  a 2 a  a

  1  2 

  2    3.)

  . . .   1   

  1

  1

  1

  1 2  2    a a : a  

  

  2

  2

  2 2  a  a a  a

    

  1  1 

  1  x y  xy  4.)  . . .

    1  1  x  y

   

  24 2 

  3 5.)   75  . . .

  2 2 

  3

  1

  10

  4

  

6

6.) 0, 25 1, 44 x10 22,5 10 243 15    . . .

27 III. RASIONALKAN :

  1

  1

  7 1.)  2.)  3. 

  3

  3

  3 1 

2 

  3 16  12 

  9 7 4 3 

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

  1

  3

  5

  3

  7 a.

  b.

  c.

  d.

  e. )

  4

  4 _{x y 2}

  4

  2

  4 (  ) 2 2. Jika xy = 7, maka nilai  . . . ₂

  ( x y  ) _{2 7}

  2 _{14 28 196}

  a. 2

  b. 2

  c. 2

  d). 2

  e. 2 x x

• – 3

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

  3

  9

  9 b.

  d).

  e.

  a. 3√3 √3

  c. 81√3

  2

  2

  4

  2   ^{2 2} x x x x

  1

  1 4. Jika a  ( e  e ) dan b  ( e  e ) maka nilai a  b  . . .

  2

  2  

   x  2x 2x

  2

  2

  a. e

  b. e

  c. e  e

  d) 1

  e. 0

  2

  49

  2

  3

  3 5. 169 3    8 12   

  64 8  50 

  13

  16

  5 a.

  b.

  c. 5

  d. 17

  e) 24

• – 29 – 11 1 x

  2 2 

  6. Nilai x yang memenuhi persamaan :  _{x } 3 adalah :

  3

  

7

  27

  a) 2,5

  b. 2

  c. 1

d.

  e. _x – 2,5 – 1,25

  2

  4

  7. Nilai x yang memenuhi :  adalah : _x

  8

  4

  2

  15

  13

  11

  9

  7 a.

  b.

  c.

  

d.

  e.     

  2

  2

  2

  2

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

  

LOGARITHMS

c b = a a > 0, a 1 

  

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

  a logb=c log b=c the logarithm of b to the base a. We write this as , sometimes as . _{c a} a

  In general: b = a  logb = c , a > 0, a ≠ 0

  1

  1

  2 10 

  3

  2

  100 10 log 100 2 or log 100

  2 2 log

  3 For example,        

  8

  8 Exercise:

  1. Convert the following to logarithm form:

  1

  4  2 q

  3 

  81 7 p  r a.

  b.  c.

  49

  2. Convert the following to exponential form:

  2 3 p q r

  a. log

  32 

  5

  b. log

  9 

  2

  c. log 

  3. Find the value of each of the following: ₁

  2 ₂

  3

  7

  a. log

  64

  b. log

  4

  c. log

  1

  d. log

  7

  8

  3

  81

  2

  2

  e. log ,

  25

  f. log( 

  9 )

  g. log

  9

  h. log

  32

  x _{x }

  1 2 log

  5

  1

  4. Find x: a. log

  64 

  1

  b. = 1

  5

  

x

  5 log

  5 

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

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

  1 

  a

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

  log

  1

  2

  d. logarithms of negative numbers are not defined, for example 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: _{a b} log a b

  1.  _{a n b m m} log _n a b

  2.   

_{a a a}

  log b log c log b . c

  3.   b

a a a

  log b log c log

  4.   _{a n a} c

  log b  n log b 5. _{a n a m n} log b  log b 6. _m

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

  6. Find the value of each the following: _{4 5 3 125 2} log 25 log 2 log 4 log 6 log

  8

  4 a. _{27 625} b.

  5 ₂ c.

  9 d.

  25 ₉ e. ₁

  8 ₈

  log 5 log 7 log log

  6 log

  10 ²

  1

  3 g.

  4 h.

  5 5 i.

  81 3 j.

  2 f.

         

  4

  7. Simplify and evaluate: _{2 2 2}

  4

  4 _{5 5 5}

  log 200 log 25 log 5 log 8 log 250

  a. log 25 + log 4 b. 

  c.  

  1

  1

  3

d. 2 log 2 + 2 log 3 + log 5 + log 7 5 .

  25

• log 49
• – log 9 + log 10 + log

  3

  2

  3

  3

  3

  3

  3

  3

  1

  1

  log 5 log 7 . log 9 log 10 log 14 . log 144

  e.     

  3

  2

  8. Expand to a single logarithm:

  2

  2

   

  x y    

  . y x y 

  3

  3

  4

  3

  2

    

  b. x

c.

  d. x  x y log log . log log .

  5    a.

   

  2

  2

  2

     

   

  z z x  y

       

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

  4

  5

  3

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

  f. log

  g. log 3 .

  6

  3

  24

  10. Evaluate: _{3 9} log

  2

  2

  2

  6 8 . log 512 log 2 log

  5 

  log

  4

  27

  a.  ₃ b.

  c. log

  2

  3 . log 2 log

  16 5 . log

  4 5 log

  8    log ,

  4

  2  x  a

  2

  2 a

  6

  log m log x : y 11. a. If  , find .

   

  y _{b b}  

  

3

  5

  3

  log y : x  n log x : y

b. If , find  

  Laws of logarithms: 7. a b b p p a

  4 , find

  8

c.

  81 log

  2 b.

  3 log

  4 , express the following in a: a.

  3 log

  17. Given a 

  1 ,

.

  25 , 1 log

   5 log

  1

  e. Given a

  3 .

  8 log

  16 , find

   27 log

  d. If m

  9 .

  5 log

  25 , find

  27 log

  9

  16

  2 ,

.

  10 log

  3 log

  6 . Express

   20 log

  6 and n

   30 log

  19. Given m

  p

  36 log

  p

c.

  p b.

  log

  1 log

  1

  2

  a.

  1 log . Express the following in a, b or c.

  2

  

  and c

p

    , 30 log 5 log

  18. Given b a ^{p p}

  c. Given a 

  125 , log

  log log log

  1

  81 log

  14. Evaluate: a.

     

   

    ^{4 3 2} log log log b d c ^{a c b} a

  log 64 log 27 log b.

  2

  5

  3

  5

  1

  13. Simplify: a.

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

  1 log 

  log

  a b b a

  log log  10.

  b b ⁿ

   log log log

9.

^{n a a}

  c c b ^{a b a}

  

8.

  1 81 log

  18 ₂ ¹

  5 , find

  5

  8 log

  b. Given p 

  243 .

  5 5 log

  27 , find

  5 log

  16. a. If p 

      .

  25

  10 log ^{25 25 5 5}

  

b.

  1

18 log

8 log 3 log

  15. Simplify:  

  

   

  1 _{4 1}

  2

  1

  1 25 log 25 log

  2

  5 log

  6 in m or n.

  2 log

  3 20. Simplify .

  5

  log 9  log

  9

  3

  25

  

15

a b

  21. If log

  5  and log 8  , find log 750 . _{b b}

  log M log a x a M 22. For a, b and M are greater than 1, and  , find x. _{ab ab a b} log log c abc bc 23. Prove :    . c b

  1 log b c log c b

  

     

  24. Given 1    . Prove a + b = c.

   a 2 b c b

  log a log a log a

  2 y 25. Given x y  a and log x  b . Find x .

  log log

    y

  2

  2

  

3

  48

  log 5 a , log 7 b

  26. Given   and log ₁ 5  c . Express log 98 in a, b or c.

  3 _{2 4}

  log 5  1 log

  20

  

2

9 log

  9 27. Evaluate:     .

  log 5 log 3 . log

  16

  2

  2

   log

  5

  1 _{3 2 3} log ₂

  10 _{4 9}

  log 36 log 4  log125 log125    ₃₆ . log5 5

  28. Evaluate:

_{3   }

_{3 2 3} . log25 . log25 log12 . 144

   