Powers of i
Now we will examine an interesting property of i. When we raise it to any
positive integer power and simplify, the result is one of only four
possibilities: i, 1, i, or 1.
Look at the powers of i listed in the table.
To simplify a higher power of i, we use this fact: i^{4} = 1.
For example, letâ€™s simplify i^{10.}

i^{10} 
Use the Multiplication Property of Exponents
to write i^{10} as a product where one factor is a
power of i that is a multiple of 4.

= i^{8} Â· i^{2} 
Rewrite i^{8} in terms of i^{4}.
Replace i^{4} with 1. Replace i^{2} with 1.
Multiply.
So, i^{10} = 1. 
= (i^{4})^{2}
Â· i^{2} = (1)^{2}
Â· i^{2}
= 1 
Note:
i^{1} = i
i^{2} = 1
i^{3} = i^{2} Â· i = (1)
Â· i = i
i^{4} = i^{2} Â· i^{2}
= (1)(1) = 1
i^{5} = i^{4} Â· i^{1}
= 1 Â· i = i
i^{6} = i^{4} Â· i^{2}
= 1 Â· (1) = 1
i^{7} = i^{4} Â· i^{3}
= 1 Â· i = i
i^{8} = i^{4} Â· i^{4}
= 1 Â· 1 = 1
i^{9} = i^{4} Â· i^{4}
Â· i = 1 Â· 1
Â· i = i
The pattern repeats:
i, 1, i, 1, i, 1, i, 1, â€¦
We can follow the same process to simplify i^{27}.
Write i^{27} using a multiple of 4. Rewrite i^{24} in terms of i^{4}.
Replace i^{4} with 1.
Replace i^{3} with i.
Multiply. 
i^{27} 
= i^{24}
Â· i^{3} = (i^{4})^{6}
Â· i^{3}
= 1^{6} Â· i^{3}
= 1 Â· (i)
= i 
So, i^{27} = i.
Example 1
Simplify.
a. i^{35 }
b. i^{82 }
c. i^{20}
Solution
a. To simplify i^{35}, divide 35 by 4.
The result is 8 with remainder 3. 

i^{35} 
= (i^{4})^{8}
Â· i^{3}
= 1^{8} Â· i^{3}
= 1 Â· (i)
= i 
b. To simplify i^{82}, divide 82 by 4. The result is 20 with remainder 2. 

i^{82} 
= (i^{4})^{20}
Â· i^{2}
= 1^{20} Â· i^{2}
= 1 Â· (1)
= 1 
c. To simplify i^{20}, divide 20 by 4.
The result is 5 with remainder 0. 

i^{20} 
= (i^{4})^{5}
Â· i^{0}
= 1^{5} Â· 1
= 1 Â· 1
= 1 
