Chapter 1-Review of algebra.pdf

8/20/2019 Chapter 1-Review of algebra.pdf

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CHAPTER 1 : REVIEW OF ALGEBRA

By Ms yaya

FE1041 MATHEMATICS 1


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Review of Algebra

•Sets of real numbers

•Exponents and radicals

•

Operations with algebraic expression

•Logarithm



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Sets of real numbers

• Rational numbers are numbers that can be written in the

form

where and are integers and ≠ 0. For

example,

,

9

and

8

.

• Irrational numbers, are numbers that cannot be written

in the form

where and are integers and ≠ 0. For

example, , and 3.



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Sets of real numbers

• When a rational number is written in the decimal form the

digits after decimal point repeats itself

• Example:

• For irrational number the decimal representation is

nonrepeating.• Example



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Exponent

na

Exponent also known as index.



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Rules of exponent

nmnm x x x .1

nmnm x x x .2

mnnm x x .3



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Rules of exponent

nn x x 1

.4

nnn y x xy )( .5

n

n

x x

1 .6



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Rules of exponent

n

nn

y

x

y

x

.7

mnnm

x x .8

1 .9 0 x



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Example : exponent

Solve the following :

1255 ) 2 xa

Step 1 : Make the base the same for left and right hand side

32 55 x

Step 2 : Equate the power

32 x

2

3 x



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Example : exponent

237

149 )

2

x

xb

Step 1 : Make the base the same for left and right hand side

)23(2 7)7(2 x x

232 7)7(2 x x



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Example : exponent

237

149 )

2

x

xb

Step 2 : Equate the power

232 2 x x

0232 2 x x

0)2)(12( x x2or2

1

x



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Example : exponent

03

433 ) 1 x xc

Step 1 : Find the common index

03

4)33(3 1 x x

034

333

x

x

x3isindexcommonThe

Step 2 : Let the common index = u

u x 3Let



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Example : exponent

03

433 ) 1 x xc

Step 3 : Solve the equation

03

4

3

uu

043 uu44 u

1u

13 x

033 x

0 x



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Radical

Also known as surd

n p

number irrational:Where p



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The conjugate

The conjugate of :ba

ba


1


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Rationalise the denominator

When a radical appears in the denominator of a

fraction, we usually rationalise the denominator

by multiplying the numerator and denominator by

a)Itself or

b) Its conjugate




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Rationalise the denominator : example

2

1)a

22

2

2

2

2

2

12




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Rationalise the denominator : example

21

1)

b

2121

211

211

2121

12




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Logarithm vs exponent

If = , then = x




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Law of logarithm

= +

Example :

= −

Example :




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Law of logarithm

=

Example :

= 1

Example :



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Law of logarithm

=

Example :




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Logarithm

The common logarithm, log =

The natural logarithm, ln =




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Logarithm : Example

Without using calculator, evaluate the following :

) 32 ) 3




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Logarithm : Example

5

33log)

30log b)

6.0loga)

:evaluate

,calculator usingwithout,32.25logand59.13logGiven that

2

2

2

22

c




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Logarithm : example

6.0loga)

:evaluate


2

22

5

3log

10

6log6.0log 222

5log3log 22

32.259.1

73.0




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Logarithm : Example

30log b)

:evaluate


2

22

)253(log30log 22 2log5log3log 222

132.259.1

91.4




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Logarithm : Example

533log)

:evaluate


2

22

c

5

18log

5

33log

22

5

23log

2

2

5log2log3log 222

2

5log2log3log2 222

86.132.21)59.1(2




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Solving equation involving logarithm

52 ) xa

5log2log 1010 x

5log2log 1010 x

2log

5log

10

10 x

322.2 x

31



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)2(log5loglog3log ) 2222 x xb

)2(5log)3(log 22 x x

)2(53 x x

1053 x x

102 x

5 x



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0)3ln(2 ) xc

0)3ln(2 x

2)3ln( x

2)3(log xe

23 e x

23 e x

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Chapter 1-Review of algebra.pdf