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Exponents,
Surds and
Logarithms
NCS Mathematics
DVD Series
Outcomes for this DVD
In this DVD you will:
• Revise the exponential notation and review the index laws.
LESSON 1.
• Simplify expressions involving rational exponents.
LESSON 2.
• Simplify expressions involving surds.
LESSON 3.
• Revise the logarithmic notation and logarithm laws.
LESSON 4.
NCS Mathematics
DVD Series
The
Exponential
Notation and
Index Laws
Lesson 1
The Exponential Notation
42 2 2 2 2
We know that :
... (to factors of , , )na a a a a n a n a
nr a
4or 2 is the product of 4 factors of 2.
Exponent
Base
Power
What if exponents are not positive integers?
0
21) 3 1a
0 0
If variable bases are non-zero and is a positive integer then:
1. 1 (0 is undefined)
12. n
n
n
a
aa
2
2
1 12) 3
3 9
Examples
Index Law 1 (Multiplication)
Law 1: n m n ma a a 4 5 91) a a a
1 22) 2 .2 2n n
2 33) 2 . 2 2 8
3 2 5
4) 3 . 3 3 243
Examples
Index Law 2 (Division)
Law 2: n m n ma a a 5
5 3 2
31)
aa a
a
66 8 2
8 2
5 1 12) 5 5
5 5 25
2 1
2
23)
10 5 5
ab a b b
a b a
Examples
Index Law 3 (Exponentiation)
Law 3: m
n nma a
3
2 2 3 61) a a a
2
5 5 2 102) x x x
Examples
Index Law 4
Law 4: ( )m m mab a b
2
3 4 3 2 4 2 6 8 81) 2 2 2 64a a a a
3
3 2 3 3 2 3 9 62) 2 3 2 3 2 3
Examples
Index Law 5
Law 5:
m m
m
a a
b b
3
3 3 3 9
4 4 3 12
2 2 21)
3 3 3
2 2
2 2
2 2 42)
x x x
Examples
Example 1: Applying Exponent Laws
• Simplification using the exponent laws.
6 3
24
6 .91)
118 .4
x x
xx
36 2
4 22 2
2.3 . 3
2.3 . 2
xx
x x
6 6 6
4 8 4 2
2 .3 .3
2 .3 .2
x x x
x x x
6 4 4 2 6 6 82 .3x x x x x x 4 4 42 .3 16 3x xor
Example 2: Applying Exponent Laws
1 1
1 11
2 42)
2 2
n n
n nn n
2
2
1 1
2 2
2 2
22
n n
nn n
2 21 1 2 22n n n n n 2 1
24
• Simplification using the exponent laws.
Multiply with the reciprocal
Tutorial 1: Simplify Expressions using
the Exponent Laws
3 2 3
23 2
2 2
4
21
1
Simplify the following expressions:
2 .81)
14 .4
12 22)
8 3
18 2.33)
2
n n
n
x x
x x
x x
x
PAUSE DVD
• Do Tutorial 1
• Then View Solutions
Tutorial 1 Problem 1: Suggested Solution
3 2 3
23 2
2 .81)
14 .4
n n
n
3 2 3 9
6 4 4
2 . 2
2 . 2
n n
n
3 2 3 9 6 4 42 n n n 2
Tutorial 1 Problem 2: Suggested Solution
2 2
4
12 22)
8 3
x x
x x
22 2
3 4
3 2 2
2 3
xx
xx
2 4 2 3 2 42 .3x x x x x 2 2 9
2 .34
2 4 2 2
3 4
2 .3 .2
2 .3
x x x
x x
Tutorial 1 Problem 3: Suggested Solution
2
1
1
18 2.33)
2
x x
x
2 1 2 2 22 .3x x x x 72
2 2 2 2
1
2 3 2 3
2
x x x
x
3 22 3
NCS Mathematics
DVD Series
Lesson 2
Rational
Exponents
What is a rational exponent?
Rational Number is a real number
which can be written in the form
where , and 0m
m n nn
We will in this part consider powers
where the exponent is a rational number.
We will consider expressions like m
na
Equivalent Notations
1
( , 2, )nna a n n 1
3 35 5 (From right to left)
1
554 4 (From left to right)
Equivalent Notations for negative
rational exponents
1 1
1 ( , 2,)nn na a a n n
1
3 13 31
5 5 (From left to right)5
1
5 1 551
= 4 =4 (From right to left)4
Another Equivalent Notation
2
3 232 2 (From left to right)
( 0, ; , 2) m
n mna a r a n m n
5
54 43 3 (From right to left)
• Factorize all bases into prime factors.
• Apply the exponential laws.
21341) 81 27
1 2
4 34 33 3
1 23 3
1 13
3
1 1
6 42) 125 25
1 1
3 26 45 5
1 1
2 25 5
05 1
Simplification (Without a calculator) of
single term exponential expressions
• Without variables – simplify each term and add.
32 2
3 4 33 4 35 2 2
2 3 23 341) 125 16 8
2 3 25 2 2
3125 8 324 4
Example 1: Simplification of polynomial
exponential expressions
• Without variables – simplify each term and add.
• With variables – factorize.
12
2
9 32)
3
xx
x
2 12
2
3 3 .3
3 .3
xx
x
13 3 .3
3 .9
x x
x
13 13
3 .9
x
x
23
9
2 1 2
3 9 27
Example 2: Simplification of polynomial
exponential expressions
• Without variables – simplify each term and add.
• With variables – factorize.
2
1
3.2 4.23)
2 2
m m
m m
2 3 16
2 1 2
m
m
1313
1
Example 3: Simplification of polynomial
exponential expressions
• Without variables – simplify each term and add.
• With variables – factorize.
22 2 64)
2 2
x x
x
2 2 2 3
2 2
x x
x
2
2
Note: Replace 2 with
2 2 6
6
2 3
2 2 2 3
x
x x
x x
a
a a
a a
2 3x
Example 4: Simplification of polynomial
exponential expressions
• Without variables – simplify each term and add.
• With variables – factorize.
11 12 6.2
5) 5.4
x x x
x
1
2
2 2 3
5.2
x x
x
1
2
5.2
5.2
x x
x
1
2 x x1 1
22
Example 5: Simplification of polynomial
exponential expressions
Tutorial 2: Working with exponents
PAUSE DVD
• Do Tutorial 2
• Then View Solutions
3
2 23
1 02
223
0,125
5 415
3 27
1) Simplify without calculators:
4 (a)
9
(b)
2 2
4
2 1
23
2 2
1
2)
12 2
8 3
4 2
2 2 2
5 .2
10 10 .2
x x
x x
x x
x x x
a a
a a
Simplify:
(a)
(b)
(c)
3
2 230,125
1a) Simplify, without using a calculator:
4
9
Tutorial 2 Problem 1(a): Suggested solution
3 22 32 3
2 1
3 2
3 22 1
3 2
28 82 4
27 27
Tutorial 2 Problem 1(b): Suggested solution
1 02
223
5 415
3 27
1(b) Simplify (No calculators) :
2 22
3 3
1 1 1
3 .5 3 53
2 2 2 2
1 1 1
3 .5 3 3 5
2 1
1 5
3 .5 9
2 2
4
12 2
8 3
x x
x x
2a) Simplify:
Tutorial 2 Problem 2(a): Suggested solution
2
2 2
3 4
2 .3 2
2 3
xx
x x
2 4 2 2
3 4
2 .3 .2
2 .3
x x x
x x
2 4 2 3 2 42 .3x x x x x
2 2 92 .3
4
2 1
23
4 2
2 2 2
x x
x x x
2b) Simplify:
Tutorial 2 Problem 2(b): Suggested solution
2 2 1
2 2 3
2 2
2 2
x x
x x
2
2
2 1 2
2 1 8
x
x
1 1
7 7
Tutorial 2 Problem 2(c): Suggested solution
2 2
1
5 5 .2 .2
5.2 5.2 .2
a a
a a
2 2
1
5 .2
10 10 .2
a a
a a
2(c) Simplify:
2 2
1 1
5 5 .2 .2
5 .2 5 .2 .2
a a
a a a a
2 2
1 1
5 5 .2 .2
5 .2 (1 5 .2 .2)
a a
a a
2 25 2
115
4
4 5 1254 25 4 5
5
NCS Mathematics
DVD Series
Lesson 3
Surds
We Define
1
Thus n na a
,
.
If then
where and
nn a x x a
n a
0 0.
NOTE
If is even, we must have and n a x
Definition of Surds
11
3 33 31) 8 8 2 2
1
1 5 55 52) 32 32 2 2
11
4 44 43) 81 81 3 3
1
1 2 22 24) 16 16 16 4
16 has no meaning (Imaginary numbers)
2Note: a a
Examples directly from the Definition
Use definition
to check!
Property 1: n n na b ab
Applications of the Multiplication Property
1) 2 3 6 (Left to right)
2) 12 4 3 2 3 (Right to left)
Multiplication Property for Surds
3 63) 10 10 (From left to right)
Property 2: m n mna a
3 36 24) 16 16 4 (From right to left)
Composition Property for Surds
Applications of the Composition Property
Property 3: n
m nm a a
Application of the Exponent Property
3 3 344 45) 2 2 8a a a
Exponent Property for Surds
Property 4: n
nn
a a
bb
Applications of the Division Property
2 16) (From left to right)
36
9 9 37) (From right to left)
4 24
Division Property for Surds
1) 2 8 4 32 3 50
2 4 2 4 16 2 3 25 2
4 2 16 2 15 2
3 2
Example 1: Simplification of surd expressions
without using a calculator
2) 50 18 32
5 2 3 2 4 2
5 2 7 2
35 2 70
3 3 3Note: also a a a a a a a
Example 2: Simplification of surd expressions
without using a calculator
2 2
3) 2 3 2 3
4 4 3 3 4 4 3 3
14
Example 3: Simplification of surd expressions
without using a calculator
1 14)
2 8
1 1
2 2 2
2 1
2 2
3
2 2
3 2
2 2 2
3 2
4
Example 4: Simplification of surd expressions
without using a calculator
Moving the surd from the
denominator to the numerator
1 15)
3 2 3 2
3 2 3 2
3 2 3 2
4
3 4
4
Example 5: Simplification of surd expressions
without using a calculator
1
1 16)
8 2 8 3 2
1
1 1
3 2 5 2
15 3
15 2
15 2
2
Example 6: Simplification of surd expressions
without using a calculator
5 11. 127)
45. 33
5 11.2 3
3 5. 3 11
10
3 5 10 5
15
2 5
3
Example 7: Simplification of surd expressions
without using a calculator
12 32
Simplify, without a calculator:
(1)
Tutorial 3: Surds
PAUSE DVD
• Do Tutorial 3
• Then View Solutions
1
1 1
12 3 12 3 3
(3)
3 12 27
7 3 75
(2)
12 32
Simplify, without a calculator:
(1)
Tutorial 3 Example 1: Suggested solution
Easier option?
12 2 12 3 3
15 2 36
15 2 6 27
Tutorial 3 Example 2: Suggested solution
6 3 3 3
7 3 5 3
3 12 27
7 3 75
(2) Simplify without a calculator
3 3 1412 3
Tutorial 3 Example 3: Suggested solution
1
1 1
12 3 12 3 3
(3) Simplify without calculator
1
1 1
2 3 3 2 3 3 3
1
1 1
3 3 5 3
1
5 3
15 3
15 3
2
NCS Mathematics
DVD Series
Lesson 4
Logarithms
If then logy
ax a y x
Exponential form Logarithmic form
log log (zero and negative number)
Both undefined
a a
Example
Definition Logarithms
22 or logyx y x
Note:
0
1 (Trivial)
0
a
a
x
21) From log 8 x
3x
22) From log 5p 5to 2p
32p
3) From log 25 2b 2
2 1to 255
b
15
b
3to 2 8 2x Basis the same
Exponents the same
Exponents the same
Basis the same
Changing from Exponential form to Logarithmic
form and visa verse
log 1 if 0 and 1 m m m m 1m m
log 1 0 if 0 and 1t t t 0 1t
10 log logx x
No base indicated That the base is 10.
log100 2
log0,01 2
log1 0, log10 1,
1 log0,1 log10 1,
General Remarks
log log log a a aAB A BLaw 1:
log log log a a aA A B
BLaw 2:
log logr
a aP r P Law 3:
loglog
log s
a
s
PP
aLaw 4:
Let ,A B
Logarithmic Laws
Change Basis
log log log a a aAB A BLaw 1:
log5 log2 log10 1
log200 log2 log100 log2 2
Applications of Logarithmic Law 1
log log log a a aA A B
BLaw 2:
2 3log 2 log3 log6 log log1 0
6
Applications of Logarithmic Law 2
log logr
a aP r P Law 3: 2 35 2
2log5 3log 2 log 2 log log1000 32
Applications of Logarithmic Law 3
120 3 log120 log3x x
log3 log120x
log120
4.35 Use 78 l
Calog
c3
ulatorx
loglog
log s
a
s
PP
aLaw 4:
25
log125 3log5 3log 125
log 25 2log5 2
2 24
2 2
log 8 3log 2 3log 8
log 4 2log 2 2
Application of Logarithmic Law 4
log x
a a x
4
3log 3 44
2 2log 16 log 2 4
3
5 51log log 5 3125
4
4
1 12 2
1log 2 log 42
log loglog
log log
xx
a
a x aa x
a a
Useful Logarithm Hint
2 3 71) log 8 log 27 log 7 3 3 1 5
2) 2log60 2log2 2log3 2
2 2
60log
2 3
3600log
36 log100 2
0,23) log 125
log125
log 0,2
3
1
log5
log5 3
Simplification of Logarithmic expressions
without using a calculator
8 3 3 814) log 16 2log log 1 log 0,25
9
1log2 2 log34log 2 403log 2 log3 log8
2 log 244
3 3log 2
2
4 4 45) 5log 2 log 0,125 2log 8 5
4 3 6
2log
2 2
4log 4 1
log9 log 46)
log8 log 27
2log3 2log 2
3log 2 3log3
2 log3 log 2
3 log3 log 2
2
3
Simplification of Logarithmic expressions
without using a calculator
5 7 12
3
1)
3 2log 2log log12
4 5
log 125 log 49 - 2log 144
log16 - log9
2log 2 log3
log 125 log 25
3log 5
x x
x
Evaluate, without a calculator:
(a)
(b)
(c)
(d)
2) log 2 , log3 , log 7 .
, .
log36
log5
12log
7
a b c
a b c
Given
Express the following in terms
of and
(a)
(b)
(c)
Tutorial 4: Logarithms
PAUSE DVD
• Do Tutorial 4
• Then View Solutions
Suggested solutions
Tutorial 4: Logarithms
1) Evaluate, without a calculator:
3 2 3 4 1 1 (a) log 2log log12 log log 2
4 5 4 25 12 100
5 7 12
3log5 2log7 4log12(b) log 125 log 49 2log 144 3 2 4 1
log5 log7 log12
2 2log 2 log3log16 log9 4log 2 2log3(c) 2
2log 2 log3 2log 2 log3 2log 2 log3
3 723log 5 log 5 log 5log 125 log 25 3 3 7 2 14(d) 3 3 93 33log 5 log 5 log 5
2 2
x x xx x
x x x
2) log 2 , log3 log7 .
, .
a b c
a b c
Given and
Express the following in terms of and
Suggested solutions
Tutorial 4: Logarithms (continued)
2 2log36 log 2 3 2log 2 2log3 2 2a b (a)
10log5 log log10 log 2 12
a (b)
212 1 1log log12 log 7 log 2 .3 log 72 27
1 2log 2 log3 log 72
1 22 2 2
b ca b c a
(c)
End of the DVD on Exponents, Surds
and Logarithms
REMEMBER!
•Consult text-books for additional examples.
•Attempt as many as possible other similar examples
on your own.
•Compare your methods with those that were
discussed in the DVD.
•Repeat this procedure until you are confident.
•Do not forget:
Practice makes perfect!
END