What is an Exponent?

Exponents

The repeated addition of numbers can be written in short form (product form).
Examples:

Also, we can write the repeated multiplication of numbers in a short form known as exponential form.
For example, when 5 is multiplied by itself for two times, we write the product 5 × 5 in exponential form as 5² which is read as 5 raised to the power two.

Similarly, if we multiply 5 by itself for 6 times, the product 5 × 5 × 5 × 5 × 5 × 5 is written in exponential form as 5⁶ which is read as 5 raised to the power 6.
In 5⁶, the number 5 is called the base of 5⁶ and 6 is called the exponent of the base.

In general, we write,
An exponential number as b^a, where b is the base and a is the exponent.
The notation of writing the multiplication of a number by itself several times is called the exponential notation or power notation.

Thus, in general we find that :
If ‘a’ is a rational number then ‘n’ times the product of ‘a’ by itself is given as a × a × a × a ….. , n times and is denoted by aⁿ, where ‘a’ is called the base and n is called the exponent of aⁿ.

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Examples

1. Write the following statements as repeated multiplication and complete the table:

S.No.	Statements	Repeated Multiplication	Short form
(i)	3 multiplied by 3 for 6 times	3 × 3 × 3 × 3 × 3 × 3 = 729	3⁶
(ii)	2 multiplied by 2 for 3 times	2 × 2 × 2	2³
(iii)	1 multiplied by 1 for 7 times	1 × 1 × 1 × 1 × 1 × 1 × 1	1⁷

2. Write the base and exponent of following numbers. And also write in expanded form:

S.No.	Numbers	Base	Exponent	Expanded Form	Value
(i)	3⁴	3	4	3 × 3 × 3 × 3	81
(ii)	2⁵	2	5	2 × 2 × 2 × 2 × 2	32
(iii)	3³	3	3	3 × 3 × 3	27
(iv)	2²	2	2	2 × 2	4
(v)	1⁷	1	7	1 × 1 × 1 × 1 × 1 × 1 × 1	1

Exponents of Negative Integers

When the exponent of a negative integer is odd, the resultant is a negative number, and when the power of a negative number is even, the resultant is a positive number.When the exponent of a negative integer is odd, the resultant is a negative number, and when the power of a negative number is even, the resultant is a positive number.
or (a negative integer) an odd number = a negative integer.
(a negative integer) an even number = a positive integer.

Examples:

Ex.1 Express 144 in the powers of prime factors.
Solution:
144 = 16 × 9 = 2 × 2 × 2 × 2 × 3 × 3
Here 2 is multiplied four times and 3 is multiplied 2 times to get 144.
∴ 144 = 2⁴ × 3²

Ex.2 Which one is greater : 3⁵ or 5³ ?
Solution:
3⁵ = 3 × 3 × 3 × 3 × 3 = 9 × 9 × 3
= 81 × 3 = 243
and 5³ = 5 × 5 × 5 = 25 × 5 = 125
Clearly, 243 > 125
∴ 3⁵ > 5³

Maths