DIGITAL TEXTBOOK

Submitted by ,

GANESH KRISHNAN G

B.Ed MATHEMATICS

CLASSVIII

N.S.S. T.C PANDALAM

contentsChapter 1 EXPONENTS AND POWERS

1.1 INTRODUCTION

1.2 EXPONENTS

1.3 LAWS OF EXPONENTS

1.3.1 Multiplying Powers with the same Base

1.3.2 Dividing powers with the same base

1.3.3 Taking power of a power

1.3.4 Multiplying powers with the same

exponents

1.3.5 Dividing powers with the same

exponents

1.4 MISCELLANEOUS EXAMPLES USING THE

LAWS OF EXPONENTS

1.5 VIDEO LINK

1.6 WHAT WE HAVE DISCUSSED?

1.7 SOME RELATED WEBSITES

1.1 INTRODUCTION

Do you know what the means of earth is ? It is 5,970,000,000,000,000,000,000,000 kg! Can you read this number?

Mass of Uranus is 86,800,000,000,000,000,000,000,000 kg.

Which has greater mass, Earth or Uranus?

Distance between sun and Saturn is 1,433,500,000,000 m and distance between Saturn and Uranus is 1,439,000,000,000.can you read these numbers? Which distance is less?

These very large numbers are difficult to read, understand and compare. To make these numbers easy to read, understand and compare, we use exponents. In this chapter , we shall learn about exponents and also learn how to use them.

Chapter 1

Exponents and Powers

1.2 EXPONENTS

We can write large numbers in a shorter form using exponents.

Observe 10,000=10×10×10×10=104

The short notation 104 stands for the product 10×10×10×10 . here 10 is called the base and 4 is called the exponent.

The number 104 is read as the 10 raised to the power of 4 or simply as fourth power of 10.104 is called the exponential for of 10,000.

We can similarly express 1,000 as power of 10. Since 1,000 is 10 multiplied by itself three times.

1000=10×10×10=103

Here again 103 is the exponential form of 1,000

Similarly 1, 00,000=10×10×10×10×10=105

105 is the exponential form of 1,00,000

In both these examples , the base is 10; in case of 103 , the exponent is 3and in case of 105 the exponent is 5.

In all the above given examples, we have seen numbers whose base is 10.However the base can be any other number also.For example:

81=3×3×3×3 can be written as 81=34 , here 3 is the base and 4 is the exponent.

25 =2×2×2×2×2=32, which is the fifth power of 2.

In 25 2 is the base and 5 is the exponent.

In the same way. 243=3×3×3×3×3=35

64= 2×2×2×2×2×2=26

625=5×5×5×5=54

EXAMPLE 1 Express 256 as a power 2.

SOLUTION we have 256=2×2×2×2×2×2×2×2=28

So we can say that 256=28

EXAMPLE 2 which one is greater23 or 32 ?

SOLUTION we have ,23 =2×2×2=8,

And 32 =3×3=9

Since 9>8,so 32 is greater than 23

EXAMPLE 3 which one is greater 82 or28 ?

SOLUTION 82 =8×8=64

28 =2×2×2×2×2×2×2×2=256

Clearly, 28 >82

EXAMPLE 4 Expand a3 b2 ,a2 b3, b2 a3 ,b3 a2, are they all same?

TRY THESE

Find five more such examples, where a number is expressed in exponential form.Also identify the base and the exponent in each case.

TRY THESE

Express:1. 729 as a power of 32. 128 as a power of 2

SOLUTION a3 b2 = a3 x b2

= (a x a x a) x (b x b)

= a x a x a x b x b

a2 b3 =a2 x b3

=a x a x b x b x b

b2 a3 =b2 x a3

=b x b x a x a x a

b3 a2 =b3 x a2

=b x b x b x a x a

EXAMPLE 5 Express the following numbers as a product of powers of prime factors:

(1) 72 (2) 432 (3) 1000 (4) 16000

SOLUTION

(1) 72 = 2x36 = 2x2x18 2 72

= 2x2x2x9 2 36

= 2x2x2x3x3 2 18

= 23x32 3 9

3

Thus , 72 =23x32 (required prime factor product form)

(2) 432 = 2x216=2x2x108=2x2x2x54

= 2x2x2x2x27=2x2x2x2x3x9=2x2x2x2x3x3x3

Or 432 =24 x 33 (required form)

(3) 1000 = 2 x 500 = 2 x 2 x 250 = 2 x 2 x 2 x 125

= 2 x 2 x 2 x 5 x 25 = 2 x 2 x 2 x 5 x 5 x 5

Or 1000 = 23 x 53

(4) 16,000 = 16x 1000 =(2x2x2x2)x1000= 24 x103 (as 16 =2x2x2x2)=(2x2x2x2)x(2x2x2x5x5x5)=24 x 23 x 53 (since 1000=2x2x2x5x5x5)=(2x2x2x2x2x2x2)x(5x5x5)

Or 16000=27x53

EXAMPLE 6 Work out (1)5,(-1)3,(-1)4, (-10)3,(-5)4

SOLUTION

(1) We have (1)5=1x1x1x1x1=1In fact, you will realize that 1 raised to any powers is 1.

(2) (-1)3 =(-1)x(-1)x(-1)=1x(-1)=-1(3) (-1)4=(-1)x(-1)x(-1)x(-1)=1x1=1

You may check that (-1) raised to any odd power is (-1),And (-1)raised to any even power is (+1)

(4) (-10)3=(-10)x(-10) x(-10)=100x(-10)= -1000

(5) (-5)4=(-5)x(-5)x(-5)x(-5)=25x25=625

1. Find the value of (i) 26 (ii) 93 (iii) 112 (iv) 54

2. Express the following in exponential form:(i) 6 x 6 x 6 x 6 (ii) t x t (iii) b x b x b x b(iv) 5 x 5 x 7 x 7 x 7 (v) 2 x 2 x a x a (vi) a x a x a x c x c x c x c x d

3. Express each of the following numbers using exponential notation:(i) 512 (ii) 343 (iii) 729 (iv) 3125

4. Identify the greater number, wherever possible, in each of the following?(i) 43 or 34 (ii) 53 or 35 (iii) 28 or 82 (iv) 1002 or 2100 (v) 210 or 102

5. Express each of the following as product of powers of their prime factors:(i) 648 (ii) 405 (iii) 540 (iv) 3600

6. Simplify :(i) 2 x 10 (ii) 72 x 22(iii) 23 x 5 (iv)3 x 44

(v)0 x 102 (vi) 52 x 33 (vii) 24 x 32 (viii) 32 x 104

7. Simplify :

(-1)odd number = -1

(-1)even number = +1

Exercise 1.1

(i) (-4)3 (ii) (-3)x(-2)3 (iii) (-3)2 x (-5)2

1.3 LAWS OF EXPONENTS

1.3.1 Multiplying Powers with the Same Base (i) Let us calculate 22 x 23

22 x 23 = (2x2) x (2x2x2) = 2x2x2x2x2 =25=22+3

Note that the base in 22 and 23 is same and the sum of the exponents, i.e., 2 and 3 is 5(ii) (-3)4 x (-3)3= [(-3) x (-3) x (-3) x (-3)] x [(-3) x (-3) x (-3)]

= (-3) x (-3) x (-3) x (-3) x (-3) x (-3) x (-3) = (-3)7

= (-3)4+3

Again, note that the base is same and the sum of exponents, 4 and 3, is 7

(iii) a2 x a4= (a x a ) x (a x a x ax a)= a x a x a x a x a x a = a6

(Note: the base is the same and the sum of exponents is 2+4 = 6)

From this we can generalize that for any non-zero integer a, where m and n are whole numbers,

1.3.2 Dividing Powers with the Same Base

Let us simplify 37 ÷ 34 ?

37 ÷ 34 = 37/34=3 x 3 x 3 x 3 x 3 x 3 x 3 / 3 x 3 x 3 x 3

= 3 x 3 x 3 = 33 = 3 7-4

Thus 37 ÷ 34 = 3 7-4

Similarly,

56 ÷ 52 = 56 / 52 =5 x 5 x 5 x 5 x 5 x 5 / 5 x 5

= 5 x 5 x 5 x 5 = 54 = 5 6-2

Or 56 ÷ 52 = 5 6-2

Let a be a non-zero integer, then,

a4 ÷a2=a4/a2 =a x a x a x a / a x a = a x a = a2 = a4-2

Or a4 ÷a2 = a4-2

In general, for any non-zero integer a,

am x an = am+n

am ÷ an = am-n

TRY THESE

Simplify and write in exponential formi) 25x23 ii) p3x p2 iii)43x 42 iv) a3xa2xa7

TRY THESE

Simplify and write in exponential form:1. 29 ÷23

2. 108÷104

3. 911÷97

4. 4

Where m and n are whole numbers and m>n.

1.3.3 Taking Power of a Power

Consider the following

Simplify (23)2 ; (32)4

Now,(23)2 means 23 is multiplied two times with itself.

(23) = 23 x 23

= 23+3(Since am x an = am+n)

= 26 = 23x2

Thus (23)2=23x2

Similarly (32)4 = 32x 32x 32x 32

= 32+2+2+2

= 38 (Observe 8 is the product of 2 and 4)

= 32x4

Can you tell what would (72)10 would be equal to ?

So (23)2 = 23 x 2 = 26

(32)4 = 32x4 = 38

(72)10 = 72 x10 =720

(a2)3 = a2 x 3 = a6

(am)3 = amx3 = a3m

From this we can generalize for any non-zero integer ‘a’, where ‘m’ and ‘n’ are whole numbers,

(am)n = amn

1.3.4 Multiplying Powers With The Same Exponents

Can you simplify 23x33? Notice that here the two terms 23 and 33 have different bases, but the same exponents.

Now, 23 x 33 = (2x2x2)x(3x3x3)

= (2x3)x(2x3)x(2x3)

= 6x6x6

= 63 (Observe 6 is the product of bases 2 and 3)

Consider 44 x 34 = (4 x 4 x 4 x 4) x (3 x 3 x 3 x 3)

= (4 x 3) x (4 x 3) x (4 x 3) x (4 x 3)

= 12 x 12 x 12 x 12

= 124

Similarly, a4 x b4 = (a x a x a x a) x (b x b x b x b)

= (a x b) x (a x b) x (a x b) x (a x b)

= (a x b)4 (Note a x b =ab)

In general, for any non-zero integer a

am x bm= (ab)m

TRY THESE

Simplify and write the answer in exponential form:i.(62)4 ii.(22)100 iii. (750)2 iv. (53)7

TRY THESE

Put into another form using

am x bn= (ab)m.

i.43 x 23

ii.25 x b5

iii.a2 x t2

iv.56x(-2)6

v.(-2)4x(-3)4

(Where m is any whole number)

EXAMPLE 8. Express the following terms in the exponential form:

(i). (2 x 3)5 (ii). (2a)4 (iii). (-4m)3

SOLUTION

(i) (2 x 3)5 = (2 x 3) x (2 x 3) x(2 x 3) x(2 x 3) x(2 x 3) = (2 x 2 x 2 x 2 x 2 x 2) x (3 x 3 x 3 x 3 x 3) = 25 x 35

(ii) (2a)4 = 2a x 2a x 2a x 2a = (2 x 2 x 2 x 2) x (a x a x a x a) = 24 x a4

(iii) (-4m)2 = (- 4 x m)3

= (-4 x m) x (- 4 x m) x (- 4 x m) = (- 4) x (- 4) x (- 4) x (m x m x m) = (-4)3 x

(m)3

1. 3.5 Dividing Powers with the same ExponentsObserve the following simplifications:

(i) 24/34 = (2x2x2x2) ÷ (3x3x3x3) = (2/3) x (2/3) x (2/3) x(2/3) = (2/3)4

(ii) a3/b3 = a x a x a / b x b x b= a/b x a/b x a/b = (a/b)3

TRY THESE

Put into another form using am ÷ bm=(a/b)m

1. 45÷35

2. 25÷b5

3. (-2)3÷b3

4. p4÷q4

5. 56÷(-2)6

From these examples we may generalise

am ÷ bm = am/bm =(a/b)m where a and b are any non integers and m is a whole number.

EXAMPLE 9 Expand: (i) (3/4)4 (ii) (4/7)5

SOLUTION(i) (3/5)4 = 34/54 = (3 x 3 x 3 x 3) / (5 x 5 x 5 x 5)(ii) (4/7)5 = (-4)5/75 =((-4) x (-4) x (-4) x (-4)) / (7 x

7 x 7 x 7 x 7 )

Number with exponent zeroCan you tell what 35/35 equals to ?

35/35 = 3 x 3 x 3 x 3 x 3 /3 x 3 x 3 x 3 x 3 = 1

By using laws of exponents35 ÷ 35 = 35-5=30

So 30 = 1Can you tell what 70 is equal to ?

73 ÷ 73 =73-3 =70

And 73 ÷ 73= (7 x 7 x7)/(7x7x7) = 1

Therefore 70 = 1Similarly a3 ÷ a3 = a3-3 = a0

And a3 ÷ a3 = a 3/ a3 = a x a x a / a x a x a= 1

Thus a0 =1(for any non zero integer a)

Put into another form using am ÷ bm=(a/b)m

1. 45÷35

2. 25÷b5

3. (-2)3÷b3

4. p4÷q4

5. 56÷(-2)6

So we can say that any number(except 0) raised to the power(or exponent) 0 is 1.

1. 4 MISCELLANEOUS EXAMPLES USING THE LAWS OF EXPONENTS

Let us solve some examples using rules of exponents developed.

EXAMPLE 10 Write exponential form for 8 x 8 x 8 x 8 taking base as 2.

SOLUTION:We have 8x8x8x8 =84

But we know that 8= 2 x 2 x 2 =23

Therefore 84=(23)4 =23 x 23 x 23 x 23

= 23x4 (You may also use (am)n= amn )

= 212

EXAMPLE 11 Simplify and write the answer in the exponential form.

(i) 23 x 22 x 55 (ii) (62x64) ÷ 63 (iii) [(22)3 x 36]x 56 (iv) 82 ÷ 23

SOLUTION(i) 23 x 22 x 55=23+2 x 55

= 25 x 55=(2 x 5)5=105

(ii) (62x64) ÷ 63 = 62+4 ÷ 63

= 66/63= 66-3= 63

(iii) [(22)3 x 36]x 56 =[26 x 36] x 56

= (2x3)6 x 56

= (2x3x5)6 = 306

(iv) 8 = 2 x 2 x 2=23

Therefore 82 ÷ 23= (23)2 ÷ 23

= 26 ÷23=26-3=23

Check this Video

WHAT HAVE WE DISCUSSED ?1. Very large numbers are difficult to read,

understand , compare and operate upon. To make all these easier, we use exponents, converting many of the large numbers in a shorter form.

2. The following are exponential forms of some numbers?10,000 =104(read as 10 raised to 4)243 =35 , 128 =27.Here, 10 ,3 and 2 are the bases, whereas 4,5 and 7

are their respective exponents . we also say, 10,000 is the 4th power of 10, 243 is the 5th power of 3, etc…3. Numbers in exponential form obey cetain laws,

which are:For any non zero integers a and b and whole numbers m and n ,(a) am x an = am+n

(b) am ÷ an =am-n, m>n(c) (am)n=amn

(d) amxbm=(ab)m

(e) am ÷ bm=(a/b)m

(f) a0=1

(g) (-1)even number =1(h) (-1)odd number= -1

Some Related Websites

http://www.purplemath.com/modules/exponent.htm

http://www.mathsisfun.com/exponent.html

http://mentorminutes.com/

http://www.mathgoodies.com/lessons/vol3/exponents.html

http://www.platinumgmat.com/