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Powers of the imaginary unit

Learn how to simplify any power of the imaginary unit i. For example, simplify i²⁷ as -i.
We know that i=1 and that i2=1.
But what about i3? i4? Other integer powers of i? How can we evaluate these?

Finding i3 and i4

The properties of exponents can help us here! In fact, when calculating powers of i, we can apply the properties of exponents that we know to be true in the real number system, so long as the exponents are integers.
With this in mind, let's find i3 and i4.
We know that i3=i2i. But since i2=1, we see that:
Similarly i4=i2i2. Again, using the fact that i2=1, we have the following:

More powers of i

Let's keep this going! Let's find the next 4 powers of i using a similar method.
i5=i4iProperties of exponents=1iSince i4=1=ii6=i4i2Properties of exponents=1(1)Since i4=1 and i2=1=1i7=i4i3Properties of exponents=1(i)Since i4=1 and i3=i=ii8=i4i4Properties of exponents=11Since i4=1=1
The results are summarized in the table.

An emerging pattern

From the table, it appears that the powers of i cycle through the sequence of i, 1, i and 1.
Using this pattern, can we find i20? Let's try it!
The following list shows the first 20 numbers in the repeating sequence.
i, 1, i, 1, i, 1, i, 1, i, 1, i, 1, i, 1, i, 1, i, 1, i, 1
According to this logic, i20 should be equal to 1. Let's see if we can support this by using exponents. Remember, we can use the properties of exponents here just like we do with real numbers!
i20=(i4)5Properties of exponents=(1)5i4=1=1Simplify
Either way, we see that i20=1.

Larger powers of i

Suppose we now wanted to find i138. We could list the sequence i, 1, i, 1,... out to the 138th term, but this would take too much time!
Notice, however, that i4=1, i8=1, i12=1, etc., or, in other words, that i raised to a multiple of 4 is 1.
We can use this fact along with the properties of exponents to help us simplify i138.


Simplify i138.


While 138 is not a multiple of 4, the number 136 is! Let's use this to help us simplify i138.
i138=i136i2Properties of exponents=(i434)i2136=434=(i4)34i2Properties of exponents=(1)34i2i4=1=11i2=1=1
So i138=1.
Now you might ask why we chose to write i138 as i136i2.
Well, if the original exponent is not a multiple of 4, then finding the closest multiple of 4 less than it allows us to simplify the power down to i, i2, or i3 just by using the fact that i4=1.
This number is easy to find if you divide the original exponent by 4. It's just the quotient (without the remainder) times 4.

Let's practice some problems

Problem 1

Simplify i227.

Problem 2

Simplify i2016.

Problem 3

Simplify i537.

Challenge Problem

Which of the following is equivalent to i1?
Choose 1 answer:

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