CBSE Grade 7 Maths Chapter 13 - Exponents and Powers

Exponents and Powers

Introduction to Exponents and Powers

When the numbers given are very large like 54,32,00,00,000 then it is not easy to read them so we write them in the form of exponents.

Exponents make these numbers easy to read, write, understand and compare.

To write the large numbers in short form, we use exponents.

For any rational number a and positive integer n, we define

•  aⁿ as a × a × a × …… × a (n times). 
•  aⁿ is known as the nth power of a 
• It is read as ‘a raised to the power n’. 
• a is called the base
• n is called the exponent or power. 
  

For example:

10,000 = 10 × 10 × 10 × 10 = 10⁴

• Here, ‘10’ is called the base and ‘4’ the exponent/power/index. 
• The number 10⁴ is read as 10 raised to the power of 4 or simply as the fourth power of 10.
• 10⁴ is called the exponential form of 10,000. 

Note : 

• In general aⁿ = a x a x a x a x … n times = aⁿ
• Any number raised to power 1 gives the same number.
• Power 2 is also called square of.
• Power 3 is also called cube of.

The Expanded form of Natural Numbers

When we write the expanded form of a natural number then it can be written in exponential form.

Example

247983 = 2 × 100000 + 4 × 10000 + 7 × 1000 + 9 × 100 + 8 × 10 + 3 × 1

= 2 × 10⁵ + 4 × 10⁴ + 7 × 10³ + 9 × 10² + 8 × 10¹ + 3 × 1

Some Important Points to Remember

• (-1)^{odd number} = (-1)

• (-1)^{even number} = (1)

• a³b² ≠ a²b³

• a²b³ = b³a²

  Laws of Exponents

  1. How to multiply powers with the same base?

  If we have to multiply the powers which have the same base then we have to add the exponents.

  a^m × aⁿ = a^{m + n}

  where m and n are whole numbers and a (≠ 0) is an integer.

   Example

  8³ × 8⁴ = 8^{3 + 4} = 8⁷

  2. How to divide powers with the same base?

  If we have to divide the powers which have the same base then we have to subtract the exponents.

     

  where m and n are whole numbers and a (≠ 0) is an integer.

  Example

  
  

3. How to take the power of a power?

If we have to take the power of a power then we have to multiply the exponents.

(a^m)ⁿ = a^mn

where m and n are whole numbers and a (≠ 0) is an integer.

Example

(8³)⁴ = 8^{3 × 4} = 8¹²

4. How to multiply the powers with the same exponents?

If we have to multiply the powers where the base is different but exponents are same then we will multiply the base.

a^mb^m = (ab)^m

where m and n are whole numbers and a ,b (≠ 0) are integers.

Example

8³4³ = (8× 4)³ = 32³

5. How to divide the powers with the same exponents?

If we have to divide the powers where the base is different but exponents are same then we will divide the base.

 

Example

 

6. Numbers with Exponent Zero

Any number with zero exponents is equal to one irrespective of the base.

a^° = 1

Example

8^° = 1

7. Numbers with Exponent One

Any number with one as the exponent is equal to the number itself.

a¹ = a

Example

8¹ = 8

8. Power with a Negative Exponent

Negative exponents can be converted into positive exponents.

 

Example

 

Miscellaneous Examples

Example: 1

 

Example: 2

 

Expressing Large Numbers in the Standard Form

If we have to write very large numbers then to make them easy to read and understand we can write them in the standard form using decimals and exponents from 1.0 to 10.0.

85 = 8.5 × 10 = 8.5 × 10¹

850 = 8.5 × 100 = 8.5 × 10²

8500 = 8.5 × 1000 = 8.5 × 10³

8500 = 8.5 × 10000 = 8.5 × 10⁴

and so on.