Last updated on July 4th, 2025
Multiplication of exponents involves merging exponential terms using certain rules. To take the product of these exponents, specific rules should be applied to the base and power.
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 ap and aq be the two terms, where ‘a’ is the base and the exponents are ‘p’ and ‘q.’
If the terms are given as ap and aq, we multiply them as:
ap × aq = 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 ap × aq = a (p+q).
According to the formula,
‘a’ = 3, where ‘p’ and ‘q’ are 2 and 3
Therefore, ap × aq = 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:
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: ap x bp. 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 ap × bp (a × b)p.
Let’s consider the below-mentioned problem:
Find the product of 22 × 42
Now substitute the values in the expression ‘ ap × bp = (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 ap and bq, their product will be (ap) × (bq).
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/ap) × (1/bq)’
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.
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 b1/2.
To write the given exponential expression into radical exponent, multiply the exponent with ½. For example, (√b)n is expressed as (b1/2)n which gives bn/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 = (b1/2)n = bn/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 = 102/2 = 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 = 53/2
(√3)5 = 35/2
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:
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.
Rules | Formula |
When the bases are the same and the exponents are different, add the exponents | am/n × ax/y = am/n + x/y For example, 22/4 × 23/9 = 21/2 + 1/3 = 25/6 |
When the bases are different, but the fractional exponents are the same, multiply the base first and then evaluate. | am/n × bm/n = (a×b)m/n For example, 23/4 × 33/4 = 63/4 |
When the exponents and the bases are different, calculate them separately. | am/n × bx/y = (am/n) × (bx/y) For example, 22/3 × 33/4 ≈ 3.64 |
We use exponents in our daily life. Given below are some real-life applications. Let’s discuss them further:
Dealing with problems involving the multiplication of exponents can be tricky and lead to mistakes. Let’s discuss them further and the solutions to avoid them.
Multiply 6^4 and 5^4
304, which is 810000.
The given terms are 64 and 54. Here the bases are different and exponents remain unchanged. Hence, multiply the bases.
64 × 54 = 304 = 30 × 30 × 30 × 30 = 810000
What will be the product when 2^4 and 2^5 are multiplied?
The result is 29, which is 512
Here, we have same bases and different exponents. So, we simply add the exponents.
24 × 25 = 2(4+5) = 29 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 512
Find the product of [ (2^4 × 3^4) × (4^1 × 4^3)]
244 will be the product
(24 × 34) = 64
(41 × 43) = 44
Therefore, (24 × 34) × (41 × 43) = 64 × 44 = 244
What will be the product of (√3)^2 and (√5)^2
15
(√3)2 ×(√5)2 = (√3 × √5)2 = (√15)2 = (15)2/2 = 15
Multiply 2^2/3 and 2^3/4
217/12
Since the bases are the same and the exponents are different, add the exponents to get the product.
22/3 × 23/3 = 2(⅔ + ¾) = 217/12
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
: She loves to read number jokes and games.