Multiplication of Exponents

Before learning more about the multiplication of exponents, you should know what exponents are.

When a number gets multiplied by itself repeatedly, the number of times it gets multiplied is the value of the exponent ‘n’. For example, if you are multiplying the number 2 four times by itself, it gets expressed as: 2 × 2 × 2 × 2 = 2⁴, which then is read as 2 raised to the power of 4.

In multiplying exponents with the same base, we keep the base the same and only the exponents get added. Let a^p and a^q be the two terms, where ‘a’ is the base and the exponents are ‘p’ and ‘q.’

If the terms are given as a^p and a^q, we multiply them as: 
a^p × a^q  = a ^(p+q)

Take a look at the given example to understand the multiplication of exponents when the base is same:

Question: What will be the product of 3² and 3³?

Explanation: Here, the base is 3 and the exponents are 2 and 3.

The formula used is a^p × a^q  = a ^(p+q).

According to the formula, 


‘a’ = 3, where ‘p’ and ‘q’ are 2 and 3 
 

Therefore, a^p × a^q  = a ^(p+q) = 3² × 3³ = 3⁽²⁺³⁾ = 3⁵ 
 

 3⁵ = 3 × 3 × 3 × 3 × 3 = 243

There are some rules to multiply exponents with different bases and they depend on these two scenarios: 

• Different bases with same exponent
   
• Different bases with different exponents

Let’s discuss them in detail:

Different bases with the same exponent

For terms with different bases and the same exponent, the expression will be in the form:  a^p x b^p. Here, ‘a’ and ‘b’ are the bases, where ‘p’ will be the exponent for both the terms. To get the product, first multiply the base and then apply the exponent. Hence, we can write a^p × b^p (a × b)^p. 

Let’s consider the below-mentioned problem: 
 

Find the product of 2² × 4²
 

Now substitute the values in the expression ‘ a^p × b^p = (a × b)p’:
 

2² × 4² = (2 × 4)² = 8²
 

Since 8² is ‘8 × 8’, we get the product as 64

Different bases with different exponents

When the given terms differ in both base and exponent, they are solved separately. If the given terms are a^p and b^q, their product will be (a^p) × (b^q).

Suppose the given terms are 4² and 5³, their product will be:
(4²) x (5³)
 

First find the values of 4² and 5³. After finding the product of each, multiply them together to obtain the final product.

4² =  4 × 4 = 16
5³ =  5 × 5 × 5 = 125

Therefore, 4² × 5³ = 16 × 125 = 2000

While multiplying terms with negative exponents, we can take the reciprocal of the base. For example, 4^-2 can be expressed as 1/4² 

There are three cases to be followed while you multiply the terms with negative exponents. Let’s discuss them in more detail.
 

Case 1: The formula to be used when we have the same bases, but different negative exponents is ‘a^-p × a^-q = 1/a^(p+q)’
  
For example, find the product of 2^-1 × 2^-2

According to the formula, we find the product as

 2^-1 × 2^-2 = ½⁽¹⁺²⁾ = 1/2³ = ⅛ = 0.125

Case 2: When the bases are different, but the negative exponents are the same,
use the formula ‘a^-p × b^-p = 1/(a × b)^p’

For example, find the product of 3^-2 and 4^-2

According to the formula, we find the product as 

3^-2 × 4^-2 = 1/(3 × 4)² = 1/(12)² = 1/144 = 0.007

Case 3: When both the bases and negative exponents are different, use the formula ‘a^-p × b^-q = (1/a^p) × (1/b^q)’

For example, multiply 2^-2 and 3^-4
According to the formula, we find the product as
 2^-2 × 3^-4 = (1/2²) × (1/3⁴) = (1/4) × (1/81) =  1/(4 × 81) = 1/324 =  0.0030

The same exponent rules apply when we have to multiply exponents with variables.

• Just add the exponents when the bases are same.
  For example, find the product of a² and a³ → a² × a³ = a⁽²⁺³⁾ = a⁵

• We should first multiply the bases when the exponents are the same, but the bases are different.
  For example, find the product of a⁵ and b⁵ →  a⁵ × b⁵ = (a × b)⁵

• If both the variables and the exponents are different, evaluate them separately.
  For example, find the product of a³ and b⁹ →  a³ × b⁹ = a³b⁹.

While multiplying the exponents with a square root, the base remains a square root and the same exponential rules apply. The ‘√’ symbol is used to express the square root of a number. Hence, √b can be written in its radical form as b^1/2.

To write the given exponential expression into radical exponent, multiply the exponent with ½.  For example, (√b)n is expressed as (b^1/2)ⁿ which gives b^n/2.

Given below are the rules to be followed while multiplying the exponents with square roots.

Rule 1: Add the exponents when both the square root bases are the same

For example, find the product of  (√3)² and (√3)³

(√3)² × (√3)³ = (√3)²⁺³ = (√3)⁵

We know that (√b)ⁿ = (b^1/2)ⁿ = b^n/2. Therefore, (√3)⁵ can be expressed as (√3)^5/2.

Rule 2: If the exponents are the same and the square root bases are different, the bases are multiplied first

For example, find the product of  (√5)² and (√2)²

(√5)² ×(√2)² = (√5×√2)² = (√10)2 = 10^2/2 = 10¹ = 10

Rule 3: If the base of the square root is different from the exponent, then they are calculated separately and then multiplied together.

For example, find the product of (√5)³ and (√3)⁵

 (√5)³ = 5^3/2

 (√3)⁵ = 3^5/2

So the product will be (√5)² ×(√3)⁵ 

Here, the base will be a fraction. For e.g., a/bⁿ, where ‘a/b’ is the base and ‘n’ is the exponent.

The rules to multiply the exponents with fractions are given below:

• Rule 1: When the bases are the same, just add the exponents.
  
  (a/b)^m x (a/b)ⁿ = (a/b)^m+n
  
  For e.g., (2/4)⁶ × (2/4)⁴
  
  (2/4)⁶ × (2/4)⁴ = (2/4)⁶⁺⁴ = (2/4)¹⁰ = (½)¹⁰

• Rule 2: When the fractional bases are different and the exponents are the same, the bases get evaluated first
  
  (a/b)ⁿ x (c/d)ⁿ = [ (a/b) x (c/d) ]ⁿ
  
  For example, find the product of (1/2)⁴ × (6/12)⁴
  
  (1/2)⁴ × (6/12)⁴ = [ (1/2) x (6/12)]⁴ = (1/4)⁴

• Rule 3: When both the fractional base and exponent are different, they are evaluated separately, that is,
  
  (a/b)^m × (c/d)ⁿ = (a^m/b^m) × (cⁿ/dⁿ)
  
  For example, find the product of (2/3)³ × (3/4)²
  
  (2/3)³ × (3/4)² = (2³/3²) × (3²/4²) = (8/9) × (9/16) = 8/16 = 1/2.

Fractional exponents refer to a term’s exponent in a fractional form. Let’s consider the term ‘am/n’. Here, ‘m/n’ is the exponent.

Check the given table to understand the rules applied while multiplying.
 

We use exponents in our daily life. Given below are some real-life applications. Let’s discuss them further: 

• Finance: To calculate the growth of an investment.
   
• Science: To calculate the population growth, speed of radioactive decay, etc.
   
• Others: To calculate the area and volume, to measure and use in architecture.