chapter 1-review of algebra.pdf
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CHAPTER 1 : REVIEW OF ALGEBRA
By Ms yaya
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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
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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
,calculator usingwithout,32.25logand59.13logGiven that
2
22
5
3log
10
6log6.0log 222
5log3log 22
32.259.1
73.0
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Logarithm : Example
30log b)
:evaluate
,calculator usingwithout,32.25logand59.13logGiven that
2
22
)253(log30log 22 2log5log3log 222
132.259.1
91.4
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Logarithm : Example
533log)
:evaluate
,calculator usingwithout,32.25logand59.13logGiven that
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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Solving equation involving logarithm
)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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Solving equation involving logarithm
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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