Math 15200 Lesson 13 Section 1.4 - Mixed Review Worksheet Page 5
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1
2
3
4
5VI
(Optional) Powers of the Imaginary Unit
There is a pattern in the powers of the imaginary unit, i .
0
4
2
2
8
i
=
1
i
=
i i
=
(
−
1
)(
−
) 1
=
1
i
=
1
1
5
4
9
i
=
i
i
=
i i
=
1
i
=
i
i
=
i
2
6
4
2
10
i
1
i
i i
1 (
)(
) 1
1
i
1
=
−
=
=
−
=
−
=
−
3
2
7
4
3
11
i
i i
1
i
i
i
i i
( 1
i
)
i
i
i
=
=
−
=
−
=
=
−
=
−
=
−
Notice: Even powers of i are either 1 or -1 and odd powers of i are either i or - i .
n
To evaluate a power of i :
i
1. Determine how many groups of 4 are in n by dividing n by 4.
n
r
2. The power can be written as
i =
i
where r is the remainder after dividing.
r
3. Simplify
i . The answer will be either 1, -1, i , or - i .
Ex 4: Evaluate each power.
49
a
)
i
=
256
b
)
i
=
5
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