Zero Power Property of Exponents
Property â€”
Zero Power Property
English Any real number, except zero, raised to the power 0 is 1.
Algebra x^{0} = 1, x ≠0
Example 17^{0} = 1
Hereâ€™s a way to understand why 17 ^{0} is 1.
Suppose we write 0 as 2  2. Then, 17^{0} = 17^{2  2}.
By the Division Property of Exponents,
Since
17^{0} = 17^{2  2} and 17^{2  2} = 1, we have 17^{0}
= 1.
Note:
This same reasoning applies no matter
what power or nonzero base we choose.
Therefore, x^{0} = 1 for x ≠ 0.
Example 1
a. Use the Zero Power Property to simplify 5^{0}.
b. Justify your answer.
Solution
a. Any real number, except zero,
raised to the power 0 is 1.

5^{0} = 1 
b. Suppose we have
We can simplify this using the
Division Property of Exponents.
But if we reduce the fraction
the result is 1.
Since
is equivalent to both 5^{0} and 1,
we conclude 5^{0} = 1. 

Example 2
Find each of the following. (Assume each variable represents a nonzero
real number).
a. (7)^{0} 
b.

c. (12x^{4}y^{5})^{0 } 
d. 2y^{0} 
e. 0^{0} 
Solution
In each case, we apply the Zero Power Property: any nonzero real number
raised to the zero power is 1.
a. The base is the real number 7. 
(7)^{0} 
= 1 
b. The base, w, represents a nonzero real number. 


c. The base, 12x^{4}y^{5}, represents a nonzero
real number. 
(12x^{4}y^{5})^{0} 
= 1 
d. Only y is raised to the power 0. 
2y^{0} = 2
Â· 1 
= 2 
e. In the Zero Power Property,
the base cannot be 0. 
0^{0} is undefined 
