Multiplying Exponents
When two numbers with exponents are multiplied, it is called multiplying exponents. The multiplication of exponents is an operation performed on exponents that comes under higher grade mathematics. Sometimes learners find it difficult to understand because of different bases, negative exponents, and noninteger exponents. This lesson will help you to understand the multiplication of exponents deeply.
To multiply such numbers, we add the exponents, which means we find the total number of times a variable or a number is multiplied by itself. Let's go ahead and learn more about multiplying exponents.
What is Multiplying Exponents?
Before exploring the concept of multiplying exponents, let us recall the meaning of exponents. An exponent can be defined as the number of times a quantity is multiplied by itself. For example, when 2 is multiplied thrice by itself, it is denoted by 2 × 2 × 2 = 2^{3}. Here, 2 is the base, and 3 is the power or exponent. It reads as "2 raised to the power of 3".
Now, let's discuss what multiplying exponents mean. When the two quantities with exponents are multiplied, it is called multiplying exponents. Let us go through the different cases with the help of examples to understand the concept better.
Rules for Multiplying Exponents with the Same Base
Consider two exponents with the same base, that is, a^{n} and a^{m}. Here, the base is a. When the exponents with the same base are multiplied, the powers are added, i.e., a^{m} × a^{n} = a^{{m+n}}
Let us explore some examples to understand how the powers are added.
Examples
1. Consider the multiplication of two exponents 2^{4} and 2^{2}. Here, the base is the same, that is, 2. According to the rule, 2^{4} × 2^{2} = 2^{(4+2)} = 2^{6} = 64.
Let us verify the answer. 2^{4} × 2^{2} = (2 × 2 × 2 × 2) × (2 × 2) = 2 × 2 × 2 × 2 × 2 × 2 = 2^{6} = 64
2. Consider the product of 10^{45} and 10^{39}. The base is the same, that is, 10. According to the rule, 10^{45} × 10^{39} = 10^{(45+39)} = 10^{84}.
Will the rule still remain the same if the bases are different? Let us see this in the next section of the page.
Rules for Multiplying Exponents with a Different Base
When two numbers or variables have different bases, then also we can easily multiply the exponents by following some basic rules. Here, we have two scenarios:
1. When the bases are different and the powers are the same.
Consider two exponents with a different base and the same power a^{n} and b^{n}. Here, the bases are a and b and the power is n. When multiplying exponents with different bases and the same powers, the bases are multiplied first. It can be written mathematically as a^{n} × b^{n }= (a × b)^{n}.
2. When the bases and powers are different.
Consider two exponents with different bases and powers a^{n} and b^{m}. Here, the bases are a and b. The powers are n and m. When the exponents with different bases and different powers are multiplied, each exponent is evaluated separately and then multiplied. It can be written mathematically as a^{n} × b^{m }= (a^{n}) × (b^{m}).
Examples
1. Consider the product of 5^{2} and 8^{2}. Here, the bases are different but the powers are the same. 5^{2} × 8^{2 }= (5 × 8)^{2} = 40^{2}= 1600.
2. Consider the product of 10^{3} and 7^{2}. Here, the bases and the powers are different. 10^{3} × 7^{2 }= 1000 × 49 = 49000.
Multiplying Negative Exponents Rules
Negative Exponents tell us how many times we have to multiply the reciprocal of the base. Multiplying negative exponents require certain rules that we are going to study in this section. It can further be divided into different cases:
Cases  Formula 

When the bases are the same  a^{n} × a^{m}= a^{(n+m)}= 1/a^{{n+m}} 
When the bases are different and the negative powers are the same.  a^{n}× b^{n}= (a × b)^{n} = 1/(a × b)^{n} 
When the bases and the negative powers are different.  a^{n} × b^{m}= (a^{n}) × (b^{m}) 
Examples
1. Consider the product of 2^{3} and 2^{9}. Here, the base is same, that is, 2. The powers are negative and different. Thus, 2^{3} × 2^{9} = 2^{(3+9)} = 2^{12} = 1/2^{12} = 1/4096 ≈ 0.000244
2. Consider the product of 6^{3} and 3^{3}. Here, the bases are different and the negative powers are the same. Thus, 6^{3} × 3^{3}= (6 × 3)^{3 }= 18^{3} = 1/18^{3 }= 1/5832 ≈ 0.0001715
3. Consider the product of 7^{2} and 6^{3}. Here, both the bases and the negative powers are different. Thus, 7^{2 }× 6^{3}= 1/7^{2}× 1/6^{3}= 1/(7^{2}× 6^{3}) ≈ 9.45 × 10^{5}
Rules for Multiplying Exponents with Fractions
If there is a fraction raised to an exponent, we need to understand it separately with the help of some rules. This section can further be divided into different cases:
Cases  Formula 

When the fractional bases are the same  (a/b)^{n} × (a/b)^{m} = (a/b)^{n+m} 
When the fractional bases are different but the powers are the same  (a/b)^{n} × (c/d)^{n} = (a/b × c/d)^{n} 
When the fractional bases and the powers are different  (a/b)^{n} × (c/d)^{m} = (a^{n} × c^{m})/(b^{n} × d^{m}) 
Let us explore some solved examples to understand better.
Examples
1. Consider the product of (2/3)^{2} and (15/8)^{2}. Here, the fractional bases are different but the powers are the same. Thus, (2/3)^{2} × (15/8)^{2} = (2/3 × 15/8)^{2} = (5/4)^{2} = 5^{2}/4^{2} = 25/16.
2. Consider the product of (2/3)^{2} and (2/3)^{5}. Here, the fractional bases are the same. (2/3)^{2} × (2/3)^{5 }= (2/3)^{2+5} = Thus, (2/3)^{7} = 2^{7}/3^{7} = 128/2187.
3. Consider the product of (3/4)^{2} and (2/3)^{3}. Here, the fractional bases and the powers are different. (3/4)^{2} × (2/3)^{3} = Thus, (3^{2} × 2^{3})/(4^{2} × 3^{3}) = 1/6.
Rules for Multiplying Exponents with Variables
In this case, the bases are variables. As we have studied in the above sections, the rules remain the same if the bases are variables.
 When the variable bases are the same, the powers are added.
 When the variable bases are different and the powers are the same, the bases are multiplied first.
 When the variable bases and the powers are different, the exponents are evaluated separately and then multiplied.
Examples
1. Consider the product of x^{4} and x^{10}. The variable base is the same, that is, x. x^{4} × x^{10}= x^{4+10} = x^{14}
2. Consider the product of x^{17} and y^{17}. The variable bases are different and the powers are the same, that is, x^{17} × y^{17}= (x × y)^{17} =(xy)^{17}
3. Consider the product of x^{8} and y^{9}. The variable bases and powers are different, that is, x^{8} × y^{9} = x^{8}y^{9}
Rules for Multiplying Exponents with Square Root
In this section, we will explore the multiplication of exponents where the bases have a square root. As we have studied in the above sections, the rules remain the same if the bases are variables.
 When the square root bases are the same, the powers are added.
 When the square root bases are different and the powers are the same, the bases are multiplied first.
 When the square root bases and the powers are different, the exponents are evaluated separately and then multiplied.
Examples
1. Consider the product of (√5)^{2} and (√5)^{7}. The square root bases are the same. Thus, (√5)^{2} × (√5)^{7} = (√5)^{2+7} = (√5)^{9} = (5)^{9/2}.
2. Consider the product of (√5)^{3} and (√7)^{3}. The square root bases are different and the powers are the same. Thus, (√5)^{3} and (√7)^{3}= (√5 ×√7)^{3} = (√5×7)^{3}= (√35)^{3} = (35)^{3/2}
3. Consider the product of (√5)^{3} and (√7)^{4}. The square root bases and the powers are different. Thus, (√5)^{3} ×(√7)^{4}= 11.18 × 49 ≈ 547.82
Multiplying Fractional Exponents Rules
When a quantity has fractional power, it is called a fractional exponent. For example, 2^{3/5} is a fractional exponent. We have the following cases:
Cases  Formula 

When the bases are the same.  a^{n/m}× a^{k/j }= a^{n/m+k/j} 
When the bases are different but the fractional powers are the same.  a^{n/m}× b^{n/m }= (a×b)^{n/m} 
When the bases and the fractional powers are different.  a^{n/m}× b^{k/j }= (a^{n/m}) × (b^{k/j}) 
Examples
1. Consider the product of 2^{1/2} and 2^{3/2}. Here, the bases are the same. Thus, 2^{1/2} × 2^{3/2}= 2^{1/2+3/2} = 2^{4/2} = 2^{2} = 4.
2. Consider the product of 2^{1/2} and 3^{1/2}. Here, the bases are different but the fractional powers are the same. Thus, 2^{1/2} × 3^{1/2}= (2×3)^{1/2} = 6^{1/2} = √6.
3. Consider the product of 4^{2/3} and 2^{1/3}. Here, the bases and the fractional powers are different. Thus, 4^{2/3} × 2^{1/3} ≈ 2.52×1.26 = 3.1752
Important Notes
 Zero raised to any power (excluding 0) is 0.
 1 raised to any power is 1.
 Any number raised to zero power is 1.
 Any exponent is a way of expressing repeated multiplication.
Multiplying Exponents Related Topics
 Exponential Equations
 Exponential Decay Formula
 Exponential Growth Formula
 Exponents
 Negative Exponent
 Fractional Exponents
 Exponent Rules
 Exponential Functions
 NonInteger Rational Exponents
 Irrational Exponents
 Exponential Terms
 Operations on Exponential Terms
 Exponents, Squares and Cubes
 Why do you subtract exponents when dividing powers with the same base?
 What is 3 to the 3rd power?
 What is 10 to the 4th power?
Solved Examples on Multiplying Exponents

Example 1: There are 4^{5} boxes in a store. Each box weighs 2^{3} pounds. Using the rules for multiplying exponents, find the total weight of the boxes.
Solution:
The weight of one box is 2^{3} pounds and there are 4^{5} boxes. Therefore, the total weight of the boxes is 2^{3}× 4^{5}= 8×1024 = 8192.

Example 2: James has a pack of 5^{3} folders. Each folder has 5^{2} sheets of paper. Find the total number of sheets using the rules for multiplying exponents.
Solution:
There are 5^{3} folders and the number of sheets in each folder is 5^{2}. The two numbers have the same base. Therefore, using the rules of multiplying exponents we can find the total number of sheets of paper. 5^{3}× 5^{2}= 5^{2+3}= 5^{5} = 3125.
FAQs on Multiplying Exponents
How does Multiplying Exponents Work?
Multiplying exponents means when two numbers with exponents are multiplied. Since there are different scenarios like different bases or different exponents, there are different rules that are applied to solve them. There are some basic rules that are used in almost all cases. For example,
 When the exponents with the same base are multiplied, the powers are added, i.e., a^{m} × a^{n} = a^{(m+n)}
 When multiplying exponents with different bases and the same powers, the bases are multiplied first. This can be written mathematically as a^{n} × b^{n }= (a × b)^{n}.
 When the exponents with different bases and different powers are multiplied, each exponent is evaluated separately and then multiplied. It can be written mathematically as a^{n} × b^{m }= (a^{n}) × (b^{m}).
Can you Multiply Exponents with Different Coefficients?
Exponents can be multiplied with different coefficients. For example, 3a^{2} × 4a^{3} = (3 × 4)×(a^{2} × a^{3}) = 12a^{5}. The coefficients are multiplied separately as shown in the example.
While Multiplying Exponents do you Add the Powers?
When exponents with the same bases are multiplied, the powers are added. For example, 3^{4} × 3^{5} = 3^{(}^{4+5)} = 3^{9}.
How to Multiply Exponents With Different Bases?
In order to multiply exponents with different bases and the same powers, the bases are multiplied and the power is written outside the brackets. a^{n} × b^{n }= (a × b)^{n}. For example, 2^{2} × 3^{2} = (2 × 3)^{2} = 6^{2} = 36. However, when we multiply exponents with different bases and different powers, each exponent is solved separately and then they are multiplied. a^{n} × b^{m }= (a^{n}) × (b^{m}). For example, 2^{2} × 5^{4} = (2)^{2 }× (5)^{4} = 4 × 625 = 2500.
What is Multiplying Exponents with the Same Base?
Multiplying exponents with the same base means when the bases are the same while the exponents are different. In this case, the base is kept common and the different powers are added, i.e., a^{m} × a^{n} = a^{(m+n)}. For example, 2^{3} × 2^{4} = 2^{(3 + 4)}= 2^{7}= 128
How do you Multiply Exponents with Parentheses?
When exponents are multiplied with parenthesis, the power outside the parenthesis is multiplied with every power inside the parenthesis. For example, (2a^{2}b^{3})^{2} = 2^{2} × a^{(2×2)} × b^{(3×2) }= 4a^{4}b^{6}.
What are the Rules of Multiplying Exponents?
There are different rules that are used in multiplying exponents. The basic rules for multiplying exponents are given below.
 When the exponents with the same base are multiplied, the powers are added, i.e., a^{m} × a^{n} = a^{(m+n)}
 When the exponents with different bases and the same powers are multiplied, then the common power is written outside the bracket, i.e., a^{n} × b^{n }= (a × b)^{n}.
 When the exponents with different bases and different powers are multiplied, each exponent is evaluated separately and then multiplied, i.e., a^{n} × b^{m }= (a^{n}) × (b^{m}).
How to Multiply Exponents with Negative Powers?
Multiplying exponents with negative powers follow the same set of rules as multiplying exponents with the same and different bases. The only difference here is that we should be careful with the addition and subtraction of integers for it. For example, 2^{3} × 2^{9} = 2^{(3+9)} = 2^{12} = 1/2^{12} = 1/4096 ≈ 0.000244