Multiplying and Dividing Exponential Terms
Exponents are used to express many numbers in a single expression. When two exponential terms with the same base are multiplied, their powers are added while the base remains the same. However, when two exponential terms having the same base are divided, their powers are subtracted.
Multiplying Exponential Terms
Multiplying exponents with the same base and different bases involves the application of identities. We generalize the properties of exponents and arrive at the identities.
Multiplying Exponents With the Same Base
Consider a^{m} × a^{n }, where 'a' is the common base and 'm' and 'n' are the exponents. When we multiply two exponential terms with the same base we get: 2^{2} × 2^{3} = (2 ×2) × (2 × 2 × 2) = 2^{5}. We will get the same result if we add the exponents. 2^{2} × 2^{3} = 2 ^{(2 + 3)} = 2^{5}. Thus we conclude that:
Note that this property will apply for the multiplication of any number of exponential terms with the same base:\({a^{{p_1}}} \times {a^{{p_2}}} \times {a^{{p_3}}} \times ... = {a^{\left\{ {{p_1} + {p_2} + {p_3} + ...} \right\}}}\)
Multiplying Exponents With Different Base And Same Power
Let us consider multiplying exponents with different bases and the same exponent, as in the case of a^{m }× b^{m}. For example, let us take: 11^{4 }× 3^{4}
11^{4 }= 11 × 11 × 11 × 11 and 3^{4 }= 3 × 3 × 3 × 3
11^{4 }× 3^{4 }= (11 × 11 × 11 × 11) × (3 × 3 × 3 × 3)
= 11 × 3 × 11 × 3 × 11 × 3 × 11 × 3
= 33 × 33 × 33 × 33 = 33^{4}. This can also be written and solved as: 11^{4 }× 3^{4 }= (11 ×3)^{4 }= 33^{4}. Therefore, when we multiply exponents with different bases and the same exponent, we apply the identity:
Dividing Exponential Terms
Exponents With the Same Base
Consider a^{m} ÷ a^{n }, where 'a' is the common base and 'm' and 'n' are the exponents. Now, let us divide two exponential terms which have the same base: 3^{7}÷ 3^{4}.
3^{7} =3 × 3 × 3 × 3 × 3 × 3 × 3 and 3^{4 }= 3 × 3 × 3 × 3
3^{7} ÷ 3^{4} = \(\dfrac{3 × 3 × 3 × 3 × 3 × 3 × 3}{3 × 3 × 3 × 3}\)= 3^{3}
This can also be written and solved as: 3^{7 }÷ 3^{4} = 3 ^{(74)} = 3^{3}. Note that while dividing exponential terms, if the bases are the same, we find the difference of the exponents. Hence, the identity which is used here is:
Dividing Exponents With Different Base And Same Exponent
Consider a^{m} ÷ b^{m }, where the exponents have different bases and the same exponent. For example, let us solve: 12^{3} ÷ 3^{3}.
12^{3 }= 12 × 12 × 12 and 3^{3 }= 3 × 3 × 3
12^{3} ÷ 3^{3 }= (12 ÷ 3)^{3 }= 4^{3}
This can be concluded as 12^{3 }÷ 3^{3 }= (12÷3)^{3 }= 4^{3}. Hence, while dividing exponents with different bases and the same exponent, we apply the identity:
How to Multiply and Divide Fractional Exponents?
Before understanding how to multiply or divide the fractional exponents with the same base, let's see some examples of fractional exponents with the same base. \(2^ \frac{3}{2}\) , \(2^ \frac{1}{4}\) , \(2^ \frac{4}{5}\)
Multiplying Fractional Exponents With the Same Base
Consider 2^{½ }× 2^{¾ }= 2^{(½ + ¾ ) }= \(2^\frac{5}{4}\). Here, we have applied the identity a^{m} × a^{n} = a^{m+n}
Dividing Fractional Exponents With the Same Base
Consider \(3^ \frac{3}{2} \div 3^\frac{1}{2} = 3^{(\frac{3}{2}  \frac{1}{2})}\) = 3^{1} = 3. Here, we have applied the identity a^{m} ÷ a^{n} = a^{mn}
How to Multiply and Divide Exponents With Variables?
The identities which are used in numbers are also used in exponents with variables. Let us recollect them and then use them in the following examples:
 a^{m} × a^{n} = a^{m+n }
 a^{m }× b^{m }= (a ×b)^{m }
 a^{m} ÷ a^{n} = a^{mn }
 a^{m }÷ b^{m }= (a ÷ b)^{m }
Variable as the Base
Let us see how to use the identities when the base is a variable. For example, solve: y^{2} × (2y)^{3}
We will apply the identity: a^{m }× b^{m }= (a × b)^{m }, y^{2} × (2y)^{3 }= y^{2} × 2^{3} × y^{3} = 2^{3} × y^{(2+3) }= 8y^{5}
Variable as the Exponent
Let us see how to use the identities when the exponent is a variable. For example, solve:5^{(2x} ^{1)} ÷ 5^{(x + 1)}
We will apply the identity: a^{m} ÷ a^{n} = a^{mn }, we get 5^{(2x 1  x  1)} = 5^{(x 2 )}
Related Topics
Important Notes
 If the bases of the exponents are equal in any equation then exponents must be equal. a^{p }= a^{q }then p = q
 a^{0 }= 1[ since a^{m }÷ a^{m }= 1 = a^{mm }= a^{0}]
 If we multiply two exponents with the same base then their powers will add.
 If we divide two exponents with the same base then their powers will subtract.
Solved Examples

Example 1: Solve: 3^{2} × 9^{2}
Solution: Applying the identity, a^{m }× b^{m }= (a × b)^{m}
3^{2} × 9^{2 }= 27^{2 }= 729 
Example 2: Find the value of the expression 3(a^{4}b^{3})^{10 }× 5(a^{2}b^{2})^{3 }
Solution:
3(a^{4}b^{3})^{10 }× 5(a^{2}b^{2})^{3 }
= 3 × a^{40} × b^{30} × 5 × a^{6 }× b^{6 }(Applying the identity, (a^{m})^{n }= a^{mn})
= 3 × 5 × a^{40} × a^{6 }× b^{30} × b^{6 }
= 15× a^{(40+6)} × b^{(30+6)} (Applying the identity, a^{m} × a^{n} = a^{m+n })
= 15× a^{46} × b^{36} = 15 a^{46} b^{36} 
Example 3: Find the value of the expression \(\dfrac{a^{(n2)}}{a^{(2n4)}}\)
Solution:
\(\dfrac{a^{(n2)}}{a^{(2n4)}}\) = a^{(n  2 2n + 4) } (Applying the identity,a^{m} ÷ a^{n} = a^{mn} )
= a^{(n22n+4)} =^{ }a ^{(2n)}
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FAQs on Multiplying And Dividing Exponents
What are the Rules For Dividing Exponents?
To divide the numbers or variables with the same base, we apply the identity: a^{m} ÷ a^{n} = a^{mn }
To divide the numbers or variables with different bases, we apply the identity: a^{m }÷ b^{m }= (a ÷ b)^{m }
How do You Solve Exponents in Parentheses?
The exponents inside parentheses can be solved using the identity (a^{m}) ^{n} = a^{mn}. For example, (4^{2})^{3} = 4^{(2 × 3)} = 4^{6} = 4096
Can You Multiply Exponents With Different Coefficients?
Yes, we can multiply exponents with different coefficients. For example, 2y^{2 }× 3y = 2 × 3 × y^{(2+1) }= 6y
Can we Distribute Exponents Over Division?
Yes, we can distribute exponents over division. For example, (7/2) ^{3} = 7^{3} ÷ 2^{3 }= 343/8
How to Multiply and Divide Negative Exponents?
When we multiply and divide negative exponents, we follow the same identities that we use for positive exponents. For example, we use the identity: a^{m} ÷ a^{n} = a^{mn} to solve: 2^{3}/2^{4}. This will be: 2^{(3(4))} = 2^{(3 + 4)} = 2^{1} = 2. Observe the rule of simplification of integers which changes the sign after the brackets are opened. It should also be noted that the negative sign on an exponent means the reciprocal of the number. For example, 7^{3} can also be written as: 1/7^{3}.