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When the imaginary unit, i, is raised to increasingly higher powers,
a cyclic (repetitive) pattern emerges. Remember that i 2 = -1.

Repeating Pattern of Powers of i :
i0 = 1
i4 = i3 • i = (-i) i = -i2 = 1
i8 = i 4 i4 = 1 1 = 1
i1 = i
i5 = i 4 i = 1 (i) = i
i9 = i 4 i 4 i = 1 1• i = i
i2 = -1
i6 = i 4 i2 = 1 (-1) = -1
i10 = (i 4)2 • i2 = 1 (-1) = -1
i3 = i2 i = (-1) • i = -i
i7 = i 4 i3 = 1 (-i) = -i
i11 = (i 4)2 • i3 = 1 • (-i) = -i

The powers of i repeat in a definite pattern: (i, -1, -i, 1)

Powers of i
i1
i2
i3
i4
i5
i6
i7
i8
...
Simplified Form
i
-1
-i
1
i
-1
-i
1
...

Simplifying powers of i:
You will need to remember (or establish) the powers of 1 through 4 of i to obtain one cycle of the pattern. From that list of values, you can easily determine any other positive integer powers of i.

Method 1: When the exponent is greater than or equal to 5, use the fact that i 4 = 1
and the rules for working with exponents to simplify higher powers of i.
Break the power down to show the factors of four.
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bullet When raising i to any positive integer power,
the answer is always
i, -1, -i or 1.
Another way to look at the simplification:
Method 2: Divide the exponent by 4:
• if the remainder is 0, the answer is 1 (i0).
• if the remainder is 1, the answer is i (i1).
• if the remainder is 2, the answer is -1 (i2).
• if the remainder is 3, the answer is -i (i3).


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ex    Simplify i87

By Method 1:
Break down the power to show factors of 4. (84 is the largest multiple of 4)
exi1
By Method 2:
Divide the power by 4 to find the remainder.
87 ÷ 4 = 21 with remainder 3
The answer is i3 which is -i.

 

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