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Course: Algebra 2 > Unit 2
Lesson 1: The imaginary unit iPowers of the imaginary unit
Learn how to simplify any power of the imaginary unit i. For example, simplify i²⁷ as -i.
We know that and that .
But what about ? ? Other integer powers of ? How can we evaluate these?
Finding and
The properties of exponents can help us here! In fact, when calculating powers of , 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 and .
We know that . But since , we see that:
Similarly . Again, using the fact that , we have the following:
More powers of
Let's keep this going! Let's find the next powers of using a similar method.
The results are summarized in the table.
An emerging pattern
From the table, it appears that the powers of cycle through the sequence of , , and .
Using this pattern, can we find ? Let's try it!
The following list shows the first numbers in the repeating sequence.
According to this logic, should be equal to . 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!
Either way, we see that .
Larger powers of
Suppose we now wanted to find . We could list the sequence , , , ,... out to the term, but this would take too much time!
Notice, however, that , , , etc., or, in other words, that raised to a multiple of is .
We can use this fact along with the properties of exponents to help us simplify .
Example
Simplify .
Solution
While is not a multiple of , the number is! Let's use this to help us simplify .
So .
Now you might ask why we chose to write as .
Well, if the original exponent is not a multiple of , then finding the closest multiple of less than it allows us to simplify the power down to , , or just by using the fact that .
This number is easy to find if you divide the original exponent by . It's just the quotient (without the remainder) times .
Let's practice some problems
Problem 1
Problem 2
Problem 3
Challenge Problem
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