NCERT Textbook: Exponents & Powers | NCERT Textbooks (Class 6 to Class 12) - CTET & State TET PDF Download

 Page 1


MATHEMATICS 170
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 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.
11.2  EXPONENTS
We can write large numbers in a shorter form using exponents.
Observe 10, 000 = 10 × 10 × 10 × 10 = 10
4
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 10
4
 is read as 10 raised to the power of 4 or
simply as fourth power of 10. 10
4
 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 = 10
3
Here again, 10
3
 is the exponential form of 1,000.
Similarly , 1,00,000 = 10 × 10 × 10 × 10 × 10 = 10
5
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  11
2024-25
Page 2


MATHEMATICS 170
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 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.
11.2  EXPONENTS
We can write large numbers in a shorter form using exponents.
Observe 10, 000 = 10 × 10 × 10 × 10 = 10
4
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 10
4
 is read as 10 raised to the power of 4 or
simply as fourth power of 10. 10
4
 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 = 10
3
Here again, 10
3
 is the exponential form of 1,000.
Similarly , 1,00,000 = 10 × 10 × 10 × 10 × 10 = 10
5
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  11
2024-25
EXPONENTS AND POWERS 171
W e 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 ×10
3
 + 5 × 10
2
 + 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, 2
5
  = 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 = 3
5
64 = 2 × 2 × 2 × 2 × 2 × 2 = 2
6
625 = 5 × 5 × 5 × 5 = 5
4
Find five more such examples, where a number is expressed in exponential
form. Also identify the base and the exponent in each case.
Y ou 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 = a
2
  (read as ‘a squared’ or ‘a raised to the power 2’)
a × a × a = a
3 
 (read as ‘a cubed’ or ‘a raised to the power 3’)
a × a × a × a = a
4 
(read as a raised to the power 4 or the 4
th
 power of a)
..............................
a × a × a × a × a × a × a = a
7
  (read as a raised to the power 7 or the 7
th
 power of a)
and so on.
a × a × a × b × b  can be expressed as a
3
b
2
 (read as a cubed b squared)
TRY THESE
2024-25
Page 3


MATHEMATICS 170
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 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.
11.2  EXPONENTS
We can write large numbers in a shorter form using exponents.
Observe 10, 000 = 10 × 10 × 10 × 10 = 10
4
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 10
4
 is read as 10 raised to the power of 4 or
simply as fourth power of 10. 10
4
 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 = 10
3
Here again, 10
3
 is the exponential form of 1,000.
Similarly , 1,00,000 = 10 × 10 × 10 × 10 × 10 = 10
5
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  11
2024-25
EXPONENTS AND POWERS 171
W e 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 ×10
3
 + 5 × 10
2
 + 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, 2
5
  = 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 = 3
5
64 = 2 × 2 × 2 × 2 × 2 × 2 = 2
6
625 = 5 × 5 × 5 × 5 = 5
4
Find five more such examples, where a number is expressed in exponential
form. Also identify the base and the exponent in each case.
Y ou 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 = a
2
  (read as ‘a squared’ or ‘a raised to the power 2’)
a × a × a = a
3 
 (read as ‘a cubed’ or ‘a raised to the power 3’)
a × a × a × a = a
4 
(read as a raised to the power 4 or the 4
th
 power of a)
..............................
a × a × a × a × a × a × a = a
7
  (read as a raised to the power 7 or the 7
th
 power of a)
and so on.
a × a × a × b × b  can be expressed as a
3
b
2
 (read as a cubed b squared)
TRY THESE
2024-25
MATHEMATICS 172
a × a × b × b × b × b can be expressed as a
2
b
4
 (read as a
squared into b raised to the power of 4).
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  = 2
8
EXAMPLE 2Which one is greater 2
3
 or 3
2
?
SOLUTION We have, 2
3
 = 2 × 2 × 2 = 8    and
3
2
  = 3 × 3 = 9.
Since 9 > 8, so, 3
2
 is greater than 2
3
EXAMPLE 3Which one is greater 8
2
 or 2
8
?
SOLUTION 8
2
 = 8 × 8 = 64
2
8
  = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2  =  256
Clearly, 2
8
 > 8
2
EXAMPLE 4 Expand a
3 
b
2
, a
2 
b
3
, b
2 
a
3
, b
3 
a
2
. Are they all same?
SOLUTION a
3 
b
2
  = a
3 
× b
2
= (a × a × a) × (b × b)
= a × a × a × b × b
a
2 
b
3
 = a
2
 × b
3
= a × a × b × b × b
b
2
 a
3
 = b
2
 × a
3
= b × b × a × a × a
b
3
 a
2
 = b
3
 × a
2
= b × b × b × a × a
TRY THESE
Express:
(i) 729 as a power of 3
(ii) 128 as a power of 2
(iii) 343 as a power of 7
2 72
2 36
2 18
3 9
3
Note that in the case of terms a
3
 b
2
 and a
2
 b
3
 the powers of a and b are different. Thus
a
3
 b
2
 and a
2
 b
3
 are different.
On the other hand, a
3
 b
2
 and b
2
 a
3
 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, a
3
 b
2
 = a
3
 × b
2
 = b
2
 × a
3
 = b
2
 a
3
. Similarly, a
2
 b
3
 and b
3
 a
2
 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 = 2
3
 × 3
2
Thus, 72 = 2
3
 × 3
2
 (required prime factor product form)
2024-25
Page 4


MATHEMATICS 170
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 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.
11.2  EXPONENTS
We can write large numbers in a shorter form using exponents.
Observe 10, 000 = 10 × 10 × 10 × 10 = 10
4
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 10
4
 is read as 10 raised to the power of 4 or
simply as fourth power of 10. 10
4
 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 = 10
3
Here again, 10
3
 is the exponential form of 1,000.
Similarly , 1,00,000 = 10 × 10 × 10 × 10 × 10 = 10
5
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  11
2024-25
EXPONENTS AND POWERS 171
W e 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 ×10
3
 + 5 × 10
2
 + 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, 2
5
  = 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 = 3
5
64 = 2 × 2 × 2 × 2 × 2 × 2 = 2
6
625 = 5 × 5 × 5 × 5 = 5
4
Find five more such examples, where a number is expressed in exponential
form. Also identify the base and the exponent in each case.
Y ou 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 = a
2
  (read as ‘a squared’ or ‘a raised to the power 2’)
a × a × a = a
3 
 (read as ‘a cubed’ or ‘a raised to the power 3’)
a × a × a × a = a
4 
(read as a raised to the power 4 or the 4
th
 power of a)
..............................
a × a × a × a × a × a × a = a
7
  (read as a raised to the power 7 or the 7
th
 power of a)
and so on.
a × a × a × b × b  can be expressed as a
3
b
2
 (read as a cubed b squared)
TRY THESE
2024-25
MATHEMATICS 172
a × a × b × b × b × b can be expressed as a
2
b
4
 (read as a
squared into b raised to the power of 4).
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  = 2
8
EXAMPLE 2Which one is greater 2
3
 or 3
2
?
SOLUTION We have, 2
3
 = 2 × 2 × 2 = 8    and
3
2
  = 3 × 3 = 9.
Since 9 > 8, so, 3
2
 is greater than 2
3
EXAMPLE 3Which one is greater 8
2
 or 2
8
?
SOLUTION 8
2
 = 8 × 8 = 64
2
8
  = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2  =  256
Clearly, 2
8
 > 8
2
EXAMPLE 4 Expand a
3 
b
2
, a
2 
b
3
, b
2 
a
3
, b
3 
a
2
. Are they all same?
SOLUTION a
3 
b
2
  = a
3 
× b
2
= (a × a × a) × (b × b)
= a × a × a × b × b
a
2 
b
3
 = a
2
 × b
3
= a × a × b × b × b
b
2
 a
3
 = b
2
 × a
3
= b × b × a × a × a
b
3
 a
2
 = b
3
 × a
2
= b × b × b × a × a
TRY THESE
Express:
(i) 729 as a power of 3
(ii) 128 as a power of 2
(iii) 343 as a power of 7
2 72
2 36
2 18
3 9
3
Note that in the case of terms a
3
 b
2
 and a
2
 b
3
 the powers of a and b are different. Thus
a
3
 b
2
 and a
2
 b
3
 are different.
On the other hand, a
3
 b
2
 and b
2
 a
3
 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, a
3
 b
2
 = a
3
 × b
2
 = b
2
 × a
3
 = b
2
 a
3
. Similarly, a
2
 b
3
 and b
3
 a
2
 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 = 2
3
 × 3
2
Thus, 72 = 2
3
 × 3
2
 (required prime factor product form)
2024-25
EXPONENTS AND POWERS 173
(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 = 2
4
 × 3
3
(required form)
(iii) 1000 = 2 × 500 = 2 × 2 × 250 = 2 × 2 × 2 × 125
= 2 × 2 × 2 × 5 × 25 = 2 × 2 × 2 × 5 × 5 × 5
  or 1000 = 2
3
 × 5
3
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
  or 1000 = 2
3
 × 5
3
Is Atul’s method correct?
(iv) 16,000 =  16 × 1000  =  (2 × 2 × 2 × 2) ×1000 = 2
4
 ×10
3
 (as 16 = 2 × 2 × 2 × 2)
= (2 × 2 × 2 × 2) × (2 × 2 × 2 × 5 × 5 × 5) = 2
4
 × 2
3
 × 5
3
(Since 1000 = 2 × 2 × 2 × 5 × 5 × 5)
= (2 × 2 × 2 × 2 × 2 × 2 × 2 ) × (5 × 5 × 5)
  or, 16,000 = 2
7
 × 5
3
EXAMPLE 6 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
Y ou 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) 2
6
(ii) 9
3
(iii) 11
2
(iv) 5
4
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
2024-25
Page 5


MATHEMATICS 170
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 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.
11.2  EXPONENTS
We can write large numbers in a shorter form using exponents.
Observe 10, 000 = 10 × 10 × 10 × 10 = 10
4
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 10
4
 is read as 10 raised to the power of 4 or
simply as fourth power of 10. 10
4
 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 = 10
3
Here again, 10
3
 is the exponential form of 1,000.
Similarly , 1,00,000 = 10 × 10 × 10 × 10 × 10 = 10
5
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  11
2024-25
EXPONENTS AND POWERS 171
W e 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 ×10
3
 + 5 × 10
2
 + 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, 2
5
  = 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 = 3
5
64 = 2 × 2 × 2 × 2 × 2 × 2 = 2
6
625 = 5 × 5 × 5 × 5 = 5
4
Find five more such examples, where a number is expressed in exponential
form. Also identify the base and the exponent in each case.
Y ou 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 = a
2
  (read as ‘a squared’ or ‘a raised to the power 2’)
a × a × a = a
3 
 (read as ‘a cubed’ or ‘a raised to the power 3’)
a × a × a × a = a
4 
(read as a raised to the power 4 or the 4
th
 power of a)
..............................
a × a × a × a × a × a × a = a
7
  (read as a raised to the power 7 or the 7
th
 power of a)
and so on.
a × a × a × b × b  can be expressed as a
3
b
2
 (read as a cubed b squared)
TRY THESE
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MATHEMATICS 172
a × a × b × b × b × b can be expressed as a
2
b
4
 (read as a
squared into b raised to the power of 4).
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  = 2
8
EXAMPLE 2Which one is greater 2
3
 or 3
2
?
SOLUTION We have, 2
3
 = 2 × 2 × 2 = 8    and
3
2
  = 3 × 3 = 9.
Since 9 > 8, so, 3
2
 is greater than 2
3
EXAMPLE 3Which one is greater 8
2
 or 2
8
?
SOLUTION 8
2
 = 8 × 8 = 64
2
8
  = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2  =  256
Clearly, 2
8
 > 8
2
EXAMPLE 4 Expand a
3 
b
2
, a
2 
b
3
, b
2 
a
3
, b
3 
a
2
. Are they all same?
SOLUTION a
3 
b
2
  = a
3 
× b
2
= (a × a × a) × (b × b)
= a × a × a × b × b
a
2 
b
3
 = a
2
 × b
3
= a × a × b × b × b
b
2
 a
3
 = b
2
 × a
3
= b × b × a × a × a
b
3
 a
2
 = b
3
 × a
2
= b × b × b × a × a
TRY THESE
Express:
(i) 729 as a power of 3
(ii) 128 as a power of 2
(iii) 343 as a power of 7
2 72
2 36
2 18
3 9
3
Note that in the case of terms a
3
 b
2
 and a
2
 b
3
 the powers of a and b are different. Thus
a
3
 b
2
 and a
2
 b
3
 are different.
On the other hand, a
3
 b
2
 and b
2
 a
3
 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, a
3
 b
2
 = a
3
 × b
2
 = b
2
 × a
3
 = b
2
 a
3
. Similarly, a
2
 b
3
 and b
3
 a
2
 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 = 2
3
 × 3
2
Thus, 72 = 2
3
 × 3
2
 (required prime factor product form)
2024-25
EXPONENTS AND POWERS 173
(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 = 2
4
 × 3
3
(required form)
(iii) 1000 = 2 × 500 = 2 × 2 × 250 = 2 × 2 × 2 × 125
= 2 × 2 × 2 × 5 × 25 = 2 × 2 × 2 × 5 × 5 × 5
  or 1000 = 2
3
 × 5
3
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
  or 1000 = 2
3
 × 5
3
Is Atul’s method correct?
(iv) 16,000 =  16 × 1000  =  (2 × 2 × 2 × 2) ×1000 = 2
4
 ×10
3
 (as 16 = 2 × 2 × 2 × 2)
= (2 × 2 × 2 × 2) × (2 × 2 × 2 × 5 × 5 × 5) = 2
4
 × 2
3
 × 5
3
(Since 1000 = 2 × 2 × 2 × 5 × 5 × 5)
= (2 × 2 × 2 × 2 × 2 × 2 × 2 ) × (5 × 5 × 5)
  or, 16,000 = 2
7
 × 5
3
EXAMPLE 6 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
Y ou 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) 2
6
(ii) 9
3
(iii) 11
2
(iv) 5
4
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
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MATHEMATICS 174
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) 4
3
 or 3
4
(ii) 5
3
  or 3
5
(iii) 2
8
 or 8
2
(iv) 100
2
 or 2
100
(v) 2
10
 or 10
2
5. Express each of the following as product of powers of their prime factors:
(i) 648 (ii) 405 (iii) 540 (iv) 3,600
6. Simplify:
(i) 2 × 10
3
(ii) 7
2
 × 2
2
(iii) 2
3
 × 5 (iv) 3 × 4
4
(v) 0 × 10
2
(vi) 5
2
 × 3
3
(vii) 2
4
 × 3
2
(viii) 3
2
 × 10
4
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 × 10
12 
; 1.5 × 10
8
(ii) 4 × 10
14  
; 3 × 10
17
11.3  LAWS OF EXPONENTS
11.3.1  Multiplying Powers with the Same Base
(i) Let us calculate 2
2
 × 2
3
2
2
 × 2
3
 = (2 × 2) × (2 × 2 × 2)
= 2 × 2 × 2 × 2 × 2 = 2
5
 = 2
2+3
Note that the base in 2
2
 and 2
3
 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) a
2
 × a
4
 = (a × a) × (a × a × a  × a)
= a × a × a × a × a × a =  a
6
(Note: the base is the same and the sum of the exponents is 2 + 4 = 6)
Similarly , verify:
4
2
 × 4
2
 = 4
2+2
3
2
 × 3
3
 = 3
2+3
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