Exponents: Types & Properties

An exponent is a method of representing any large or small number in terms of powers. It is defined as raised to the power of a number. 

• For example, consider term 2⁴, the number 2 is the base and 4 is the exponent of the term. 
• It defines that 2 is multiplied by itself 4 times, so 2⁴ = 2 x 2 x 2 x 2. It is read as 2 raised to the power of 4.
• An exponent defines the number of times a term (or value) is multiplied by itself. 
• The symbol (^) is used to represent the exponent. For example, 2 raised to a power of 3 is written as 2^3 or 23 which equals 8. 
• An exponent can be a whole number, negative number, fraction, and decimal.

Also Read: Uses of Exponents to Express Small Numbers in Standard Form

Key Terms: Exponents, Power, Laws of Exponents, Zero Exponent, Fraction Exponent, Base, Decimal Exponent, Exponent Form

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Meaning of Exponents

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Exponent is the way of representing a large or small number in the form of power. For example, when 3 is multiplied 4 times by itself, it can be expressed as 3 x 3 x 3 x 3 = 3⁴. 

Now, consider the below expression to represent the exponent of a number. 

yⁿ = (y . y . y . …….. . y)

The above expression explains that yn means y is multiplied by itself ‘n’ times.

In the term yⁿ, 

• y is called the base
• n is the exponent or power
• yⁿ is read as ‘y to raise to n’ or ‘y to the power of n’.

A few examples of exponents are:

• 6 x 6 x 6 x 6 = 6⁴
• (-4) x (-4) x (-4) = (-4)³

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Properties of Exponents

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The properties of exponents are used to simplify complex problems involving exponents to solve them. Properties of exponents are also known as the laws of exponents. The laws of exponents are:

1. Law of Product: According to this law, when exponents with the same base are multiplied, the exponents are added. So, 

a^m x aⁿ = a^{m + n}

2. Law of Quotient: When two exponents with a common base are in the division, the exponent in the denominator gets subtracted by the numerator exponent. Thus,

aⁿ /a^m = (a)^{n - m}

3. Law of Zero Exponent: It says if the exponent of any real number is zero, then it is equal to 1. 

a⁰ = 1

4. Law of Negative Exponent: According to this law, when an exponent is negative, it can be converted to a positive exponent by reciprocating the base. It is given as,

a^-n = 1/aⁿ, where a < 0 (non-zero number) and n is a real number. 

5. Law of Power of Power: This law states that, when a power of base is raised to another power, the exponents are multiplied. It is represented as:

(an)^m = a^nm

6. Law of Power of Product: When two different bases with the same exponent (power) are multiplied, then it is equal to the product of bases raised to the common exponent. It is given as:

a^m b^m = (ab)^m

7. Law of Power of Quotient: When two different bases with the same exponent are in the division, it can be represented as

a^m/ b^m = (a/b)^m 

Also Read: Powers and Negative Exponents

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Types of Exponents

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According to the number in the power of the term, the exponent can be classified into 5 types. 

• Positive Exponent
• Negative Exponent
• Zero Exponent
• Rational or Fraction Exponent
• Decimal Exponent

Also Read: Multiplication and Division of Integers

Positive Exponents

A positive exponent is defined as an exponent with a power greater than zero. 

• It is written when a base is multiplied by itself many times.
•  For example, the expression an says that a is multiplied by itself n times where n > 0.
• Some examples of positive exponents are 4⁵, 21⁷, (-6)⁴. 

Negative Exponents

If the exponent of the base is less than zero then the exponent is called a negative exponent. 

• It tells the number of times the reciprocal of the base is multiplied by itself. 
• For expression a-m, it can be written as 1/am which tells the reciprocal of a i.e., 1/a is multiplied by itself ‘m’ times. 
• Some examples of negative exponents are 7^-3 = 1/7³ , 2x 6^-21 = 2/6²¹ , (-12)^-4 = 1/(-12)⁴

Zero Exponents

An expression with an exponent equal to 0 is called a zero exponent expression and it is equal to 1 whatever the value of base is. For example, 

• 5⁰ = 1
• (100)⁰ = 1
• 1/(20)⁰ = 1/ 1 = 1 

Rational (or Fraction) Exponents

If the exponent of a base is a rational (or fraction) it is known as a rational exponent. Square root, cube root, and nth root are examples of rational exponents. 

• A number having power 1/2 is called the square root of a base.
• A number having power 1/3 is called the cube root of a base.
• Similarly, a number with power 1/n is called the n^th root of a base.

Some examples of rational exponents are 6^2/3, -27^1/3, and 10^5/3. They can be simplified as

• 6^2/3 = (6²)^1/3 = 36^1/3 
• -27^1/3 = ((-3)³)^1/3 = – 3
• 10^5/3 = (10⁵)^1/3 

Decimal Exponents

If the exponent of the term is in decimal form, then it is called a decimal exponent. 

• For a decimal exponent, the approximate value is considered to evaluate an answer.
• An example of a decimal exponent is 2^1.5, it can be written in a rational form as 2^3/2. Few examples of decimal exponents are : 2^0.25 = 2^1/4 ,3^0.75 = 3^¾

Also Read: Rational Expressions

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Things to Remember

• An exponent defines how many times the number is multiplied by itself. 
• Thus, 2⁵ means 2 is multiplied by itself 5 times.
• The expression aⁿ x bⁿ is written as (a x b)ⁿ. 
• If an exponent is negative, then take a reciprocal to make the exponent positive i.e., (b/a)^-n = (a/b)ⁿ.
• When exponents with the same base are in division, like b^n/m, it is written as b^{(n - m)}.
• Any term having an exponent equal to zero always has a value equal to 1. So, a⁰ = 1. 
• To solve decimal exponents, first convert it to fraction form i.e., 4^0.5 can be written as 4^1/2.

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