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MATHEMATICS170

11.1 INTRODUCTION

Do you know what the mass 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 betw een Saturn and Uranus is 1,439,000,000,000 m. Can you read these numbers? Which dis tance 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.

11.2 EXPONENTS

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

Observe10, 000 =10 × 10 × 10 × 10 = 104

The short notation 10

4 stands for the product 10×10×10×10. Here '10' is called

the base and '4' the exponent. The number 104 is read as 10 raised to the power of 4 or simply as fourth power of 10. 104 is called the exponential form of 10,000. We can similarly express 1,000 as a power of 10. Note that

1000 =10 × 10 × 10 = 103

Here again, 10

3 is the exponential form of 1,000.

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

5 is the exponential form of 1,00,000

In both these examples, the base is 10; in case of 10

3, the exponent

is 3 and in case of 10

5 the exponent is 5.Exponents and

Powers

Chapter 11Rationalised 2023-24

EXPONENTS AND POWERS171We have used numbers like 10, 100, 1000 etc., while writing numbers in an expanded form. For example, 47561 = 4 × 10000 + 7 × 1000 + 5 × 100 + 6 ×

10 + 1

This can be written as 4 × 10

4 + 7 ×103 + 5 × 102 + 6 × 10 + 1.

Try writing these numbers in the same way 172, 5642, 6374. 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 = 3

4, here 3 is the base and 4 is the exponent.

Some powers have special names. For example,

10

2, which is 10 raised to the power 2, also read as '10 squared' and

10

3, which is 10 raised to the power 3, also read as '10 cubed'.

Can you tell what 5

3 (5 cubed) means?

5

3 = 5 × 5 × 5 = 125

So, we can say 125 is the third power of 5.

What is the exponent and the base in 5

3? Similarly, 25 = 2 × 2 × 2 × 2 × 2 = 32,which is the fifth power of 2. In 2

5, 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

Find five more such examples, where a number is expressed in exponential form. Also identify the base and the exponent in each case. You can also extend this way of writing when the base is a negative integ er.

What does (-2)

3 mean?

It is(-2)3 =(-2) × (-2) × (-2) = - 8

Is(-2)4 =16?Check it.

Instead of taking a fixed number let us take any integer a as the base, and write the numbers as, a

× a =a2 (read as 'a

squared' or 'a raised to the power 2') a × a × a =a3 (read as 'a cubed' or 'a raised to the power 3') a × a × a × a =a4 (read as a raised to the power 4 or the 4th power of a) a

× a × a × a × a × a × a = a7 (read as a raised to the power 7 or the 7th power of a)

and so on. a

× a × a × b × b can be expressed as a3b2 (read as a cubed b squared)TRY THESERationalised 2023-24

MATHEMATICS172

a × a × b × b × b × b can be expressed as a2b4 (read as a squared into b raised to the power of 4). E

XAMPLE 1 Express 256 as a power 2.

SOLUTIONWe have 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2.

So we can say that 256 = 2

8

EXAMPLE 2Which one is greater 23 or 32?

SOLUTIONWe have, 23 = 2 × 2 × 2 = 8 and

3

2 = 3 × 3 = 9.

Since 9 > 8, so, 3

2 is greater than 23

EXAMPLE 3Which one is greater 82 or 28?

SOLUTION82 =8 × 8 = 64

2

8 =2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256

Clearly,28 >82

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

SOLUTIONa3 b2 =a3 × b2

= (a × a × a) × (b × b) =a × a × a × b × b a

2 b3 =a2 × b3

=a

× a × b × b × b

b

2 a3 =b2 × a3

=b × b × a × a × a b

3 a2 =b3 × a2

=b × b × b × a × aTRY THESEExpress: (i)729 as a power of 3 (ii)128 as a power of 2 (iii)343 as a power of 7272 236
218
3 9

3Note that in the case of terms a3 b2 and a2 b3 the powers of a and b are different. Thus

a

3 b2 and a2 b3 are different.

On the other hand, a3 b2 and b2 a3 are the same, since the powers of a and b in these two terms are the same. The order of factors does not matter. Thus, a3 b2 = a3 × b2 = b2 × a3 = b2 a3. Similarly, a2 b3 and b3 a2 are the same. EXAMPLE 5Express the following numbers as a product of powers of prime factors: (i)72(ii)432(iii)1000(iv)16000

SOLUTION

(i)72 =2 × 36 = 2 × 2 × 18 =2 × 2 × 2 × 9 =2 × 2 × 2 × 3 × 3 = 23 × 32 Thus,72 =23 × 32 (required prime factor product form)Rationalised 2023-24 EXPONENTS AND POWERS173(ii)432 = 2 × 216 = 2 × 2 × 108 = 2 × 2 × 2 × 54 =2 × 2 × 2 × 2 × 27 = 2 × 2 × 2 × 2 × 3 × 9 =2 × 2 × 2 × 2 × 3 × 3 × 3 or432 =24 × 33(required form) (iii)1000 = 2 × 500 = 2 × 2 × 250 = 2 × 2 × 2 × 125 =2 × 2 × 2 × 5 × 25 = 2 × 2 × 2 × 5 × 5 × 5 or1000 =23 × 53

Atul wants to solve this example in another way:

1000 =10 × 100 = 10 × 10 × 10

= (2 × 5) × (2 × 5) × (2 × 5)(Since10 = 2 × 5) =2 × 5 × 2 × 5 × 2 × 5 = 2 × 2 × 2 × 5 × 5 × 5 or1000 = 23 × 53

Is Atul's method correct?

(iv)16,000 = 16 × 1000 = (2 × 2 × 2 × 2) ×1000 = 24 ×103 (as 16 = 2 × 2 × 2 × 2)

=(2 × 2 × 2 × 2) × (2 × 2 × 2 × 5 × 5 × 5) = 24 × 23 × 53 (Since 1000 = 2 × 2 × 2 × 5 × 5 × 5) =(2 × 2 × 2 × 2 × 2 × 2 × 2 ) × (5 × 5 × 5) or,16,000 =27 × 53 EXAMPLE 6Work out (1)5, (-1)3, (-1)4, (-10)3, (-5)4.

SOLUTION

(i)We have (1)5 = 1 × 1 × 1 × 1 × 1 = 1 In fact, you will realise that 1 raised to any power is 1. (ii)(-1)3 = (-1) × (-1) × (-1) = 1 × (-1) = -1 (iii)(-1)4 = (-1) × (-1) × (-1) × (-1) = 1 ×1 = 1 You may check that (-1) raised to any odd power is (-1), and (-1) raised to any even power is (+1). (iv)(-10)3 = (-10) × (-10) × (-10) = 100 × (-10) = - 1000
(v)(-5)4 = (-5) × (-5) × (-5) × (-5) = 25 × 25 = 625

EXERCISE 11.1

1.Find the value of:

(i) 26(ii)93(iii)112(iv)54

2.Express the following in exponential form:

(i)6 × 6 × 6 × 6(ii)t × t(iii)b × b × b × b

(iv)5 × 5× 7 × 7 × 7(v)2 × 2 × a × a(vi)a × a × a × c × c × c × c × d

odd number(-1)= -1 even number(-1)= + 1

Rationalised 2023-24

MATHEMATICS174

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 factor

s: (i)648(ii)405(iii)540(iv)3,600

6.Simplify:

(i)2 × 103(ii)72 × 22(iii)23 × 5(iv)3 × 44 (v)0 × 102(vi)52 × 33(vii)24 × 32(viii)32 × 104

7.Simplify:

(i)(- 4)3(ii)(-3) × (-2)3(iii)(-3)2 × (-5)2 (iv)(-2)3 × (-10)3

8.Compare the following numbers:

(i)2.7 × 1012 ; 1.5 × 108(ii)4 × 1014 ; 3 × 1017

11.3 LAWS OF EXPONENTS

11.3.1 Multiplying Powers with the Same Base

(i)Let us calculate 22 × 23 2

2 × 23 =(2 × 2) × (2 × 2 × 2)

=2 × 2 × 2 × 2 × 2 = 25 = 22+3

Note that the base in 2

2 and 23 is same and the sum of the exponents, i.e., 2 and 3 is 5

(ii)(-3)4 × (-3)3 = [(-3) × (-3) × (-3)× (-3)] × [(-

3) × (-3) × (-3)]

=(-3) × (-3) × (-3) × (-3) × (-3) × (-3) × (-3) =(-3)7 =(-3)4+3 Again, note that the base is same and the sum of exponents, i.e., 4 and

3, is 7

(iii)a2 × a4 = (a × a) × (a × a × a × a = a × a × a × a × a × a = a6 (Note: the base is the same and the sum of the exponents is 2 + 4 = 6)

Similarly, verify:

4

2 × 42 =42+2

3

2 × 33 =32+3Rationalised 2023-24

EXPONENTS AND POWERS175Can you write the appropriate number in the box. (-11)2 × (-11)6 =(-11)b

2 × b3 =b (Remember, base is same; b is any integer).

c

3 × c4 =c (c is any integer)

d

10 × d20 = dFrom this we can generalise that for any non-zero integer a, where m

and n are whole numbers, a m × an =am + n

Caution!

Consider 2

3 × 32

Can you add the exponents? No! Do you see 'why'? The base of 2

3 is 2 and base

of 3

2 is 3. The bases are not same.

11.3.2 Dividing Powers with the Same Base

Let us simplify 3

7 ÷ 34?

quotesdbs_dbs28.pdfusesText_34

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