Sometimes, it is very difficult to read, understand and compare very large numbers.

E.g., distance between Sun and Saturn is 1,433,500,000,000 m, distance between Saturn and Uranus is 1,439,000,000,000 m.

To make these numbers easy to read, understand and compare, we use the concept of exponents.

Exponents

The concept of exponent makes the representation of large numbers easier.

Example: let a number 10,000

We can write 10,000 = 10 × 10 × 10 × 10 = 10⁴

Here the notation 10⁴ is the short notation of 10 × 10 × 10 × 10 i.e. 10, 000

So, 10⁴ is the exponential form of 10,000

Here 10 is called the base and 4 the exponent. The number 10⁴ is read as 10 raised to the power of 4 or fourth power of 10.

Similarly, 1,000 = 10 × 10 × 10 × 10 = 10³

So, 10³ is the exponent form of 1,000

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

10⁵ is the exponential form of 1,00,000

We can use numbers like 10, 100, 100, etc to write the numbers in expanded form.

Example: 

12534 

The numbers which have base other than 10 can also be represented in exponent form.

64  = 2 × 2 × 2 × 2 × 2 × 2 

Here, 2 is the base and 6 is the exponent.

Similarly, 125 = 5 × 5 × 5 = 5³

81 = 3 × 3 × 3 × 3 = 3⁴

243 = 3 × 3 × 3 × 3 × 3 = 3⁵ 

We can also write the negative numbers in exponent form.

Example: 

−27 

−32 

−125 

There are numbers which have more than one base. We can also write these numbers in exponent form.

Example: 

36 = 2 × 2 × 3 × 3 = 2² × 3³  

675 = 3 × 3 × 3 × 5 × 5 = 3³ × 5²  

1296 = 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 

Question: Express the following in exponential form:

Solution:

Question: Identify the greater number, wherever possible, in each of the following:

Solution:

Laws of Exponents

Numbers in exponential form obey certain laws, which are discribed as follows:

(1) Multiplying Powers with the Same Number

For any non−zero integer a, where m and n are whole numbers,

Example: 

(2) Dividing Powers with the Same Number

For any non−zero integer a, where m and n are whole numbers and m > n,

Example: 

(3) Taking Power of a Power:

For any non−zero integer a, where m and n are whole numbers,

Example: 

(4) Multiplying Powers with the Same Exponents

For any non−zero integer a, where m is any whole numbers,

Example: 

(5) Dividing Powers with the Same Exponents

For any non−zero integer a and b, where m is any whole numbers,

Example: 

(6) Any number (except 0) raised to the power 0 is always 1.

Example: 

Question: Simplify

Solution:

Decimal Number System

We can express a large number in terms of 10s.

Example: let a number is 35246

Here, 3 is at ten thousand place, 5 is at thousand place, 2 is at hundred palce, 4 is at 10 tenth place and 6 is at unit place. So, 35246 can be written as

35246 

Similarly, expansion of 250374 will be as follows:

250374 

Question: Write the following numbers in the expanded form:

Solution:

Expressing Large Number in The Standard Form

We know that reading, writing and understanding a large number is not convenient. So, to make it convenient, we use the concept of powers.

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.

Example: Let a number be 3500

We can express this number in many ways using exponent.

All of these are the representation of the number 3500 in exponent form. But the standard form of the number 3500 is that number which is expressed as number between 1 and 10.

So, 3500 = 3.5 × 1000 = 3.5 × 10³ is the standard form of the number 3500

Similarly, 12756000

Question: Express the following numbers in standard form:

Solution:

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