Exponents – Product Form, Exponential Notation, Positive Exponents, Negative Exponents

Exponents are shorthand for repeated multiplication of the same thing by itself. Usually, Exponent says how many times to use a number in the multiplication, In this article, we tried covering everything about the Exponents such as exponential form and product form,  positive and negative rational exponents, negative integral exponents, laws of exponents, etc. in the coming modules.

• Laws of Exponents
• Rational Exponent
• Integral Exponents of a Rational Numbers
• Solved Examples on Exponents
• Practice Test on Exponents
• Worksheet on Exponents

In general, we use exponents to represent large numbers in the short form so that it’s convenient to read. Base a raised to the power n is equal to multiplication of “a” n times. The Process of using Exponents is nothing but raised to the power where the exponent is called power. For better understanding, we even provided solved examples explaining step by step solution.

Example

4*4*4*4*4 can be written as 4⁵  and is read as 4 raised to the power of 5.

In the same way, for any rational number “a”, and a is a positive integer we define aⁿ as a x a x a x a x……a(n times)

(-a)ⁿ = aⁿ  if n is even

= -aⁿ if n is odd

Rational Number a is called the base whereas n is called exponent or power or index. In general, writing the product of writing a rational number by itself multiple times is called exponential notation or power notation.

Examples on Exponential Form

We can denote -3*-3*-3*-3 in the exponent form as (-3)⁴ and is read as -3 raised to the power of 4. in which -3 is the base.

1/2*1/2*1/2 in exponent form as (1/2)³  where 3 is the power and 1/2 is the base.

Examples on Product Form

We can denote (7)³ in the form of 7*7*7 and its product is 343.

Powers with Positive Exponents

For 5² = 5*5 = 25
5³ = 5*5*5 = 125
5⁴ = 5*5*5*5 = 625

Powers with Negative Exponents

Powers with Negative exponents is called Negative Integral Exponents.

Thus, 5^-1 = 1/5

5^-2 = 1/25
5^-3 = 1/125

Therefore, for any non zero rational number and a positive integer, we have a^-n= 1/aⁿ

a^-n is the reciprocal of aⁿ.

Solved Examples on Exponents

1. Express the following in Power Notation

(i) 1/4*1/4*1/4

= (1/4)³

(ii) (-3)*(-3)*(-3)

(-3)³

2. Express the following as Rational Numbers

(i) (2/5)³

= 2/5*2/5*2/5

= 8/125

(ii) (-2)⁴

= -2*-2*-2*-2

= 16/1

3. Express each of the following in the Exponential Form

(i) 625/81

We can write 625 = 5*5*5*5 = 5⁴

81 = 3*3*3*3 = 3⁴

Therefore, 625/81 = 5⁴/3⁴

(ii) -1/32

-1 = -1*-1*-1*-1*-1 = (-1)⁵

32= 2*2*2*2*2 = 2⁵

Therefore, -1/32 expressed in rational number as (-1)⁵/2⁵

4. Express each of the following in product form and find the value.

(i) (1/5)³
= 1/5*1/5*1/5
= 1/125

(ii) (-4)³

= -4*-4*-4

= -64