Exponents and Powers

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 between Saturn and Uranus is 1,439,000,000,000 m. 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

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’ 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, 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 3 and 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.

            Some powers have special names.

For example,

            102, which is 10 raised to the power 2, also read as ‘10 squared’ and 103, which is 10 raised to the power 3, also read as ‘10 cubed’.

            You can also extend this way of writing when the base is a negative integer.

            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 soon

                        a × a × a × b × b can be expressed as a3 b2 (read as a cubed b squared)

 

Example 1: Express 256 as a power 2.

Solution

                    We have 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2.

                    So we can say that 256 = 28

 

Example 2: Which one is greater 23 or 32 ?

Solution

                    We have, 23 = 2 × 2 × 2 = 8 and 32 = 3 × 3 = 9.

                    Since 9 > 8, so, 32 is greater than 2

 

Example 3: Expand a3 b2, a2 b3, b2 a3, b3 a2. Are they all same?

Solution

                        a3 b2 = a3 × b2

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

                                 = a × a × a × b × b

                        a2 b3 = a2 × b3

                                 = a × a × b × b × b

                        b2 a3 = b2 × a3

                                 = b × b × a × a × a

                        b3 a2 = b3 × a2

                                 = b × b × b × a × a       

Note that in the case of terms a3 b2 and a2 b3 the powers of a  and  b are different.    Thus a3 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 4: Express 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).

 

(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

                              or                   

      432 = 24 × 33

 

(iii)          1000 = 2 × 500 = 2 × 2 × 250 = 2 × 2 × 2 × 125

                       = 2 × 2 × 2 × 5 × 25 = 2 × 2 × 2 × 5 × 5 × 5

                  Or                 

1000 = 23 × 53

 

(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 5 :Work 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

 

3 Laws Of Exponents

3.1 Multiplying Powers with the Same Base

(i)   Let us calculate 22 × 23 

                        22 × 23 = (2 × 2) × (2 × 2 × 2)

                                    = 2 × 2 × 2 × 2 × 2 = 25 = 22+3

Note that the base in 22 and 23 is sa2me 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:   42 × 42 = 42+2,

                                    32 × 33 = 32+3

Caution!

Consider 23 × 32 Can you add the exponents? No! Do you see ‘why’? The base of 23 is 2 and base of 32 is 3. The bases are not same.so,we can’t add exponents.

 

3.2 Dividing Powers with the Same Base

Let us simplify 37 ÷ 34 ?

                                   

                                               

Thus,                        

(Note, in 37 and 34 the base is same and 37 ÷ 34 becomes 37-4)

Let a be a non-zero integer, then,

                                   

                                               

Thus,                        

In general, for any non-zero integer a,

            am ÷ an = am-n

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

 

3.3 Taking Power of a Power

Consider the following

Simplify         

Now,  means 23 is multiplied two times with itself.

                                   

                                                   (Since am × an = am+n)

                                               

Thus                                     

 

Similarly                 

                                   

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

                                   

 

3.4 Multiplying Powers with the Same Exponents

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

Now,              23 × 33 = (2 × 2 × 2) × (3 × 3 × 3)

                                     = (2 × 3) × (2 × 3) × (2 × 3)

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

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

                                    = (4 × 3) × (4 × 3) × (4 × 3) × (4 × 3)

                                    = 12 × 12 × 12 × 12

                                    = 124

Consider, also,       32 × a2   = (3 × 3) × (a × a)

                                    = (3 × a) × (3 × a)

                                    = (3 × a)2

                                    = (3a)2    (Note: 3×a = 3a)

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

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

                                    = (a × b)4

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

In general, for any non-zero integer a

                                    am × bm  = (ab)m    

 

Example 1: Express the following terms in the exponential form:

                        (i) (2 × 3)5   (ii) (2a)4       (iii) (– 4m)3

Solution

(i)               (2 × 3)5 = (2 × 3) × (2 × 3) × (2 × 3) × (2 × 3) × (2 × 3)

            = (2 × 2 × 2 × 2 × 2) × (3 × 3× 3 × 3 × 3)

            = 25 × 35

 

(ii)            (2a)4 = 2a × 2a × 2a × 2a

        = (2 × 2 × 2 × 2) × (a × a × a × a)

        = 24 × a4

 

(iii)          (– 4m)3 = (– 4 × m)3 = (– 4 × m) × (– 4 × m) × (– 4 × m)

            = (– 4) × (– 4) × (– 4) × (m × m × m)

            = (– 4)3 × (m)3

 

3.5 Dividing Powers with the Same Exponents

 

Observe the following simplifications:

(i)  

 

(ii)

 

From these examples we may generalise

           

Where a and b are any non-zero integers and m is a whole number.

 

Example 1: Expand: (i)     (ii)

Solution

(i)               

(ii)             

 

 

 

v Numbers with exponent zero

Can you tell what equals to?

                       

 

By using laws of exponents

                       

So,                 

 

                       

 

Similarly     

And               

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

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

 

Example 1:   Write exponential form for 8 × 8 × 8 × 8 taking base as 2.

Solution

                     We have,   8 × 8 × 8 × 8 = 84

But we know that

                                    8 = 2 × 2 × 2 = 23

Therefore               

                        84 = (23 )4

                                    = 23 × 23 × 23 × 23

Example 1:   Simplify and write the answer in the exponential form.

            (i)   (ii) 23 × 22 × 55      (iii) (62 × 64 ) ÷ 63            (iv) [(22 )3 × 36 ] × 56      (v) 82 ÷ 23

Solution

(i)                

                  = (37-2) ×35 = 35 ×35

                  = 35+5 = 310

 

(ii)            23 × 22 × 55

                  = 23+2 × 55 = 25 × 55

                  = (2 × 5)5 = 105

 

(iii)          (62 × 64 ) ÷ 63

                  66-3 = 63

 

(iv)          [(22 )3 × 36 ] × 56        

                  = [26 × 36] × 56

                  = (2 × 3)6 × 56

                  = (2×3 ×5)6

                  = 306

(v)            82 ÷ 23 

                  82= 2 × 2 × 2 = 23

Therefore 82 ÷ 23 = (23)2 ÷ 23 = 26 ÷ 23 = 26-3 = 23

 

Example 2: Simplify:

            (i)    (ii) 23 × a3 × 5a4        

Solution

(i)               We have

                   

                             

                             

                              = 210-9 × 310-6 = 21 ×34

                              = 2 × 81 = 162

 

(ii)            23 × a3 × 5a4

                  = 23 × a3 × 5 × a4

                  = 23 × 5 × a3 × a4

                  = 8 × 5 × a3+4

                  = 40 a7

 

5 Decimal Number System

Let us look at the expansion of 47561, which we already know:

            47561 = 4 × 10000 + 7 × 1000 + 5 × 100 + 6 × 10 + 1

We can express it using powers of 10 in the exponent form:

Therefore,  

                        47561 = 4 × 104 + 7 × 103 + 5 × 102 + 6 × 101 + 1 × 100

                        (Note 10,000 = 104, 1000 = 103, 100 = 102, 10 = 101 and 1 = 100

Let us expand another number:

                        104278 = 1 × 100,000 + 0 × 10,000 + 4 × 1000 + 2 × 100 + 7 × 10 + 8 × 1

                                    = 1 × 105 + 0 × 104 + 4 × 103 + 2 × 102 + 7 × 101 + 8 × 100

                                    = 1 × 105 + 4 × 103 + 2 × 102 + 7 × 101 + 8 × 100

Notice how the exponents of 10 start from a maximum value of 5 and go on decreasing by 1 at a step from the left to the right upto 0.

 

 

6 Expressing Large Numbers in the Standard Form

Let us now go back to the beginning of the chapter. We said that large numbers can be conveniently expressed using exponents. We have not as yet shown this. We shall do so now.

1.     Sun is located 300,000,000,000,000,000,000 m from the centre of our Milky Way Galaxy. 

2.     Number of stars in our Galaxy is 100,000,000,000.

3.     Mass of the Earth is 5,976,000,000,000,000,000,000,000 kg.

 

These numbers are not convenient to write and read. To make it convenient we use powers.

Observe the following:

            59 = 5.9 × 10 = 5.9 × 101

            590 = 5.9 × 100 = 5.9 × 102

            5900 = 5.9 × 1000 = 5.9 × 103

            59000 = 5.9 × 10000 = 5.9 × 104 and so on.

 

We have expressed all these numbers in the standard form. Any number can be expressed as a decimal number between 1.0 and 10.0 including 1.0 multiplied by a power of 10. Such a form of a number is called its standard form. Thus,

            5,985 = 5.985 × 1,000 = 5.985 × 103 is the standard form of 5,985.

Note, 5,985 can also be expressed as 59.85 × 100 or 59.85 × 102. But these are not the standard forms, of 5,985. Similarly, 5,985 = 0.5985 × 10,000 = 0.5985 × 104 is also not the standard form of 5,985.

We are now ready to express the large numbers we came across at the beginning of the chapter in this form.

            The, distance of Sun from the centre of our Galaxy i.e.,

                                    300,000,000,000,000,000,000 m can be written as

3.0           × 100,000,000,000,000,000,000 = 3.0 × 1020 m

 

Example 1 Express the following numbers in the standard form:

            (i) 5985.3    (ii) 65,950 (iii) 3,430,000       (iv) 70,040,000,000

Solution

(i)               5985.3 = 5.9853 × 1000 = 5.9853 × 103

(ii)            65,950 = 6.595 × 10,000 = 6.595 × 104

(iii)          3,430,000 = 3.43 × 1,000,000 = 3.43 × 106

(iv)          70,040,000,000 = 7.004 × 10,000,000,000 = 7.004 × 1010

 

A point to remember is that one less than the digit count (number of digits) to the left of the decimal point in a given number is the exponent of 10 in the standard form. Thus, in 70,040,000,000 there is no decimal point shown; we assume it to be at the (right) end. From there, the count of the places (digits) to the left is 11. The exponent of 10 in the standard form is 11 – 1 = 10. In 5985.3 there are 4 digits to the left of the decimal point and hence the exponent of 10 in the standard form is 4 – 1 = 3.

 

Summary

Numbers in exponential form obey certain laws, which are: For any non-zero integers a and b and whole numbers m and n,

a)          am ÷ an = am-n

b)         

c)           am × bm  = (ab)m

d)         

e)           a0 = 1

f)             (–1)even number = 1

(–1) odd number = – 1