1

1

Unit 1

Chapter-3

Partial Fractions, Algebraic Relationships, Surds, Indices, Logarithms

3.1 Partial Fractions:

A fraction of the form 1

733

2

x

x where the degree of the numerator is less than the degree of

the denominator is referred to as a proper fraction. If the degree of the numerator is greater

than or equal to the degree of the denominator the fraction is referred to as an improper

fraction.

An improper fraction of the form 1

12

2

x

xmay be written as

1

21

1

2

1

1

1

1222

2

2

2

xxx

x

x

x

This process is called expressing or decomposing a single fraction as a sum of two (or

several) separate fractions or Partial fractions.

Example1

Consider the rational expression 11

22

x

x

x.

This expression can be expressed in the compact form as 11

1122

2

xx

xxx= 11

232

2

xx

xx.

Example 2

Consider the expression 42

3

xx

x .

This expression can be expressed as the sum of two Partial fractions as

42

3

xx

x= 42

x

B

x

A.

Here the values of A and B are to be determined in the following way,

42

3

xx

x= 42

x

B

x

A.

243 xBxAx

2

2

Substituting 2x gives 6

5A and substituting 4x gives

6

1B . So

42

3

xx

x=

46

1

26

5

xx.

3.1.1 Supplementary Problems

1))1)(1(

3

xx 2)

)1)(4(

3

xx 3)

)2)(2(

1

xx

x

4) )2)(12(

3

xx 5)

xx

x

)1(

3

6)

)23)(1(

12

xx

x

7) )3)(2)(1(

3

xxx

x 8)

)1)(3(2

422

xxx

xx 9)

)12)(13(

12

xx

x

10) )12)(1(

5 2

xxx

xx 11)

)2)(12(

31

xx

x 12)

)2)(1(

)1(

xx

xx

13) )3)(1(

42

xxx

x

Multiple choice Exercise

Type 1

1) If ,3

2

1))(1(

xx

q

cxx

px the value of p and q are:

(a) 1,2 qp (b) 1,2 qp (c) 2,1 qp (d) 1,1 qp (e) 1,1 qp

2. If 062 pxx has equal roots and pp ,0 is :

(a) 48 (b)0 (c) 6 (d)3 (e) 24

3. If 1)(4 22 qxpxx , the values of p and q are:

(a) 2,5 qp (b) 2,1 qp (c) 5,2 qp (d) 5,1 qp (e) 1,0 qp

3.2 Algebraic Relationships

Solving a quadratic equation 02 cbxax .

3

3

Let 02

22 a

cx

a

bx

0442

22

2

2

22

a

c

a

b

a

bx

a

bx

a

c

a

b

a

bx

2

22

42

2

2

4

4

2 a

acb

a

bx

a

acb

a

bx

2

4

2

2 .

Hence the two values for x are a

acb

a

bx

2

4

2

2 and

a

acb

a

bx

2

4

2

2

Note: i) when 042 acb , x has two distinct roots.

ii) when 042 acb , x has two equal roots.

iii) when 042 acb , x has no real roots.

Example:

Solve; 0162 x

162 x

16 x

142

x

14 x

But a1 , when a is a real number.

Note:

The number 1 is not a real number. It is considered as an imaginary number and denoted

with the letter i. i.e. 1i . So, in the above example the solutions of the given equation

are ix 4 and ix 4 .

4

4

3.2.1 Solve the following quadratic equations by completing the square

1) 0462 2 xx 2) 0842 xx 3) 0372 2 xx

4) 022 axx 5) 022 baxx 6) 02 cbxax

Determine the nature of the roots of the following equation but do not solve the equations.

7) 0962 xx 8) 01062 xx 9) 0352 2 xx

10) 0243 2 xx 11) 09124 2 xx 12) 09124 2 xx

13) For what values of K is 169 2 kxx a perfect square?

14) The roots of 0123 2 kxx are equal .Find K.

15)Find a if 052 axx has equal roots.

3.3 Surds

3.3.1 General rules of Surds

A number which can be expressed as a fraction of integers (assuming the denominator is

never 0) is called a rational number. Examples of rational numbers are 5

4,

2

5 and 2. A

number which cannot be expressed as a fraction of two integers is called an irrational

number. Example of irrational numbers are 3 7,2 and Π. An irrational number involving a

root is called a Surd.

3.3.2 Multiplication of Surds:

baba

For example:

i. 636123123

ii. 864232232

3.3.3 Division of Surds:

b

aba

For example:

5

5

i. 6362

72272

ii. 395

45545

3.3.4 Addition and Subtraction of Surds:

baba

baba

Example (1):

Simplify (a) 75212243 and (b) 32850

(a)

317

3103239

325234381

325234381

75212243

(b)

211

242225

21624225

32850

6

6

3.3.5 Rationalization of the Denominator

For example: 4

4

)()())((

37)3()7()37)(37(

59)5()9()59)(59(

22

22

22

nmnmnmnm

Example (2):

Simplify 3

5

3

35

3

3

3

5

3

5

Example (3):

Simplify 37

4

3737

)37(4

37

37

37

4

37

4

Example (4)

Find, without using tables or calculators, the value of 23

1

23

1

7

6

29

2323

)23)(23(

)23()23(

23

1

23

1

3.3.6 Supplementary Problems

1. Simplify each of the following:

i. 27

ii. 243

iii. 9818

iv. 11217528

v. 81255

vi. 27327

vii. 28

22463112

viii. 21

112

ix. 73

1

73

1

7

7

2. Express the following in the form of cba :

i. 2)23(

ii. 2)323(

iii. 2)13(

iv. 3223

3223

v. 2332

2332

vi. 27

27

3. Rationalize the denominators of the following, giving your answer in the

simplest form possible:

i. 35

21

35

21

ii. 3253

5237

iii. 27

14

4. Find the value of each of the following:

i. 53

1

53

1

ii. 832

1850

iii. 22 )51(

1

)51(

1

8

8

3.4 Indices

If a positive integer a is multiplied by itself three times .We get 3a , i.e. 3aaaa .Here a

is called the base and 3, the index or power .Thus 4a means the 4th power of a, In general, na means the power of a , where n is any positive index of the positive integer a.

3.4.1 Rules of indices

There are several important rules to remember when dealing with indices.

If a, b, m and n are positive integers, then

(1) nmnm aaa e.g. 1385 333

(2) nmnm aaa e.g. 11314 555

(3) mnnm aa )( e.g.

1052 5)5(

(4) mmm baba )( e.g.

555 )23(23

(5)

m

mm

b

aba

e.g.

4

44

3

535

(6) 10 a e.g. 150

(7) n

n

aa

1

e.g. 3

3

5

15

(8) nn aa

1

e.g. 33

1

88

(9) mnn

m

aa )( e.g. 233

2

)8(8

Example 1

Evaluate: (i) 32 (ii) 3

1

8 (iii) 4

3

16 (iv) 2

3

25

i. 8

1

2

12

3

3

ii. 288 33

1

iii. 821616 33

44

3

iv. 125

1

5

1

25

1

25

125

3

3

2

32

3

9

9

Example 2

Evaluate: (i) 2

1

5

2

3

1

aaa (ii) 423ba (iii) 21

15 23 aaa

i.

30

7

30

151210

2

1

5

2

3

1

2

1

5

2

3

1

a

a

a

aaa

ii.

812

4243

423

ba

ba

ba

iii.

30

17

2

1

5

2

3

1

2

1

5

2

3

1

2

115 23

a

a

aaa

aaa

3.4.2 Solving Exponential Equations

Example 3

Solve the following exponential equations: (i) 322 x (ii) 25.04 1 x

i.

5

22

322

5

x

x

x

ii.

2

11

44

4

14

25.04

11

1

1

x

x

x

x

x

Example 4

Solve the equation: xxx 2122 332

xxxx

xxx

2122222

2122

33

332

Let xy 2

18

1

0118

0178

188

2

2

ory

yy

yy

yyy

When 8

1y

3

22

8

12

3

x

x

x

10

10

When 1y

)(12 leinadmissibx

3

32

2

1

8

12 x

Hence 3x

Example 5

If 2793 2 yx and8

142 yx

, calculate the value of x and y.

28

142

12793.

2

yx

yx

From (1):

334

33

333

333

34

34

322

yx

yx

yx

yx

From (2):

432

22

222

2

122

32

32

3

2

yx

yx

yx

yx

(3) – (4): 1

66

y

y

Substitute y = 1 into (3):

1

34

x

x

1 x and 1y

11

11

3.4.3 Supplementary Problems

1. Evaluate each of the following without using a calculator:

i. 17

ii. 017

iii. 2

3

49

iv. 3

2

8

v. 5

3

243

vi. 4

1

81

vii. 3

4

27

1

viii.

2

4

1

ix. 2

1

576

1

x. 3

4

512

xi. 13

2

48

xii. 34

1

4625

1

2. Simplify each of the following giving your answer in index form:

i. 6

1

3

1

2

1

aaa

ii. 243 aaa

iii. 64 412 aa

iv. 2

3

2

5

416

aa

v.

15

5

2

3

1

ba

vi.

24

8

3

4

1

ba

vii. aa 3

viii. aa 8 7

ix. 3 24 3 aaa

x. 9 36 2 xx aa

xi. 33

3

64

8

b

a

xii.

3

42

2

9

ba

ab

3. Solve the following equations:

i. 813 x

ii. 1255 x

iii. 832 x

iv. 8

12 x

v. 2

116 x

12

12

vi. 49

17 x

vii. 15 x

viii. 34 273 xx

ix. 634 2 xx

x. 121 842 xxx

xi. 122 2793 xxx

xii. 125

145 13 xx

4. By using appropriate substitution, or otherwise, solve the following equations:

i. 1222 22 xx

ii. 3

1333 22 tt

iii. 01292 212 xx

iv. xxx 3393 312

v. xx 3439

vi. 24225 xx

vii. 033289 1 xx

viii. xx 3246813 22

5. Solve the following pairs of simultaneous equations:

i. 3437,497 yxyx

ii. 32 375,75 abab

iii. 82,2433 52 yxyx

iv. 16

12,6255 242 yxyx

v. 16

182,27813 yxyx

13

13

3.5 Logarithms

For any number y such that xay (a>0 and a≠1), the logarithm of y to the base a is defined

to be x and is denoted by yalog .

Thus

For example,

2100log10100

481log381

10

2

3

4

Note: The logarithm of 1 to any base is 0, i.e. 01log a

The logarithm of a number to a base of the same number is 1, i.e. 1log aa

The logarithm of a negative number is not defined.

Example 1

Find the value of (i) 64log2 (ii) 3log9 (iii) 9

1log 3 (iv) 25.0log8

i. Let x64log 2

6

22

264

6

x

x

x

ii. Let x3log 9

2

1

12

33

93

21

x

x

x

x

iii. Let x9

1log 3

2

33

39

39

1

2

1

x

x

x

x

iv. Let x25.0log8

xay means that yalog =x

14

14

3

2

23

22

24

1

825.0

32

3

x

x

x

x

x

Example 2

Find the logarithm of the following to the base indicated in brackets: (i) 27 (3) (ii) 64 (8)

(iii) 1000 (10) (iv) 0.25 (2)

i. 327log

327

3

3

ii. 264log

864

8

2

iii. 31000log

101000

10

3

iv. 225.0log

225.0

2

2

15

15

3.5.1 Laws of Logarithms

Proof: (1) Let xma log and yna log

xam and

yan

Multiply m by n: yx aanm

nmyxmn

amn

aaa

yx

logloglog

(2) Let xma log and yna log

xam and

yan

Divide m by n: yx aanm

nmyxn

m

an

m

aaa

yx

logloglog

(3) Let xma log

mnxnm

am

am

a

n

a

xnn

x

loglog

Example 3

Without using tables, evaluate 5log2 2

41log-70log

35

41log 10101010 :

(1) nmmn aaa logloglog

(2) nmn

maaa logloglog

(3) mnm a

n

a loglog

16

16

2

10log2

10log

100log

5 41

270

35

41log

5log2 2

41log-70log

35

41log

10

2

10

10

2

10

10101010

Example 4

Given that 2log24log 1010 p , Calculate the value of p without using tables or

Calculators.

2log24log 1010 p

)4(log 2

10 p =2

22 104 p

2p =

4100

P= 5

Since p cannot be -5 because 10log (-5) is not defined, p=5

3.5.2 Supplementary Problems

Write the following in logarithmic from:

1. 2552 2. 1120 3. 34373

4. 3

13 1

5. 32

8

1 6. 36216

Write the following in index form:

7. 38log 2 8. 4625log 5 9. 12

1log 2

17

17

10. 01log 9 11. 236

1log 6 12. 31000log10

Solve the following equations:

13. x1log 3 14. x5

2 2log 15. x5

1log 5

16. x25.0log 4 17. x128log 2 18. x7log 7

19. x5log5

20. 2

1log 4 x 21. 3log 5 x

Simply the following logarithms:

22. 5log10 23. 64log 8 24. 3

5 5log 25. 5125log 5

26. 4

1log16 27.

x5log 5 28. 12log 3 4log 3 29. 5log 7 15log 7

30. 3 5log 6 25log 6 31. 5log3log21log 222

32. 4log310log5log2 333 33. 17log2

3

3

5log2

4

17log

17

81log

2

110101010

If 6309.2log 3 and 456.15log 3 , evaluate the following without the use of calculators or

logarithm tables:

34. 10log 3 35. 15log 3 36. 25log 3

37. 5log 3 38. 5.2log 3 39.

3

13log 3

40. 8

1log 3 41. 100log 3 42. 12log 3

43. 12.0log 3 44. 08.0log 3 45. 25log 3

48. Evaluate the following without using calculators:

(1) 5log32log4log2 101010

18

18

(2) 6log7log9log914log27log 1010101010

(3) 32log2

116log

2

38log

3

2222

(4) 6

5log3

25

24log

9

10log 444

(5) 52log91log175log 101010

(6) 105

8log4

9log

7

32log

15

4log 6666

3.5.3 Common logarithms

Example 1

Solve the Equation 7.43 x

7.43 x

Taking logarithms to base 10,

7.4lg3lg x

7.4lg3lg x

x = 3lg

7.4lg = 1.409

Example 2

Given that x2

x3 = 18 , find x correct to two decimal places .

x2x3 =18

x6 =18

Taking logarithms to base 10,

x6lg = 18lg

18lg6lg x

19

19

x =6lg

18lg = 1.61(correct to 2 decimal places).

3.5.4 Supplementary Problems

1. Solve the following equations, giving your answer correct to 3 significant figures , where

necessary :

(1) 235 37 x

(2) 74.054 x (3)

22 51005 xx

(4) 0127772 xx

Simplify 5lg6lg

125lg8lg27lg

10. Given that ba 3 ,

(1) Find the value of b)

3

1( in terms of a.

(2)Find the value of a and b if 14353 22 bb

2.4 Summary

Surds:

(1) abba (2) b

aba

(3) ba and ba are conjugate surds. The product of Conjugate surds is a rational

number.

Indices:

(1) nmnm aaa (2) nmnm aaa (3) mnnm aa )(

(4) 10 a (5) nn

aa 1

(6) nn aa

1

(7) mnn mn

m

aaa )(

Logarithms

(1) xya log means ya x .

(2) )(logloglog nmnm aaa

20

20

(3) n

mnm aaa logloglog

(4) mnm a

n

a loglog

(5) 01log a

(6) 1log aa