Multiplication Property of Exponents
Exponents have several properties. We will use these properties to simplify
expressions.
In the properties that follow, each variable represents a real number.
Multiplication Property
Property â€”
Multiplication Property of Exponents
English To multiply two exponential expressions with the same
base, add their exponents. The base stays the same.
Algebra x^{m} Â· x^{n}
= x^{m + n}
(Here, m and n are positive integers.)
Example 5^{4} Â· 5^{2} = 5^{4
+ 2} = 5^{6}
Example 1
a. Use the Multiplication Property of Exponents to simplify 2^{3}
Â· 2^{4}.
b. Use the definition of exponential notation to justify your answer.
Solution
a. The operation is multiplication and the bases are the same.
Therefore, add the exponents and use 2 as the base.
2^{3} Â· 2^{4} = 2^{3 +
4} = 2^{7}
b. Rewrite the product to show the factors. Then simplify.
Note:
Remember to add the exponents,
but leave the bases alone.
That is, 2^{3} Â· 2^{4} = 2^{3 +
4} = 2^{7} , not 4^{7}.
Note the difference between 2^{3} Â· 2^{4} and 2^{3}
+ 2^{4}.
2^{3} Â· 2^{4} = 2^{3 +
4} = 2^{7} = 128
2^{3} + 2^{4} = 8 + 16 =
24
Caution â€” Negative Bases
A negative sign is part of the base only when the negative sign is
inside the parentheses that enclose the base.
For example, consider the following cases:
In (3)^{2}, the base is 3.
(3)^{2}
= (3) Â· (3) = +9
In 3^{2}, the base is 3.
You can think of 3^{2} as the
â€œoppositeâ€ of 3^{2}.
3^{2} = (3 Â· 3) = 9
Example 2
If possible, use the Multiplication Property of Exponents to simplify each
expression:
a. ( 2)^{2} Â· ( 2)^{4}
b. 2^{2} Â· 2^{4 }
c. 2^{2} Â· ( 2)^{4}
a. In (2)^{2} Â· (2)^{4}, the base
is 2.

(2)^{2}
Â· (2)^{4} 
= (2)^{2 + 4}
= (2)^{6}
= 64 
b. In 2^{2} Â· 2^{4}, the base is 2.
We may think of 2^{2} Â· 2^{4}
as the opposite of 2^{2} Â· 2^{4}. 
2^{2}
Â· 2^{4} 
= (2^{2})
Â· (2^{4})
= (2^{2 + 4})
= (2^{6})
= 64 
c. In 2^{2} Â· (2)^{4}, the base of the first factor,
2^{2}, is 2.
The base of the second factor, (2)^{4}, is 2.
The bases are not the same, so we cannot use the Multiplication
Property of Exponents.
However, we can still evaluate
the expression. 2^{2} Â· (2^{)4}
= 4 Â· 16 = 64
