Exponents, Surds and NCS Mathematics Logarithms DVD Series resources... · 2020. 4. 8. · What is...

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