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Powers of i

The imaginary unit i is defined as the square root of −1 . So, i 2 =−1 .

i 3 can be written as ( i 2 )i , which equals −1( i ) or simply −i .

i 4 can be written as ( i 2 )( i 2 ) , which equals ( −1 )( −1 ) or 1 .

i 5 can be written as ( i 4 )i , which equals ( 1 )i or i .

Therefore, the cycle repeats every four powers, as shown in the table.

Powers of 10
i 1 =i i 0 =1
i 2 =−1 i −1 =−i
i 3 =−i i −2 =−1
i 4 =1 i −3 =i
i 5 =i i −4 =1
i 6 =−1 i −5 =−i
i 7 =−i i −6 =−1
i 8 =1 i −7 =i
i 9 =i i −8 =1
etc. etc.

 

Example 1:

Simplify.

−5 i 4

Simplify the imaginary part using the property of multiplying powers.

−5 i 4 =( −5 )⋅ i 2 ⋅ i 2

Recall the definition of i .

Since i 2 =−1 :

=( −5 )⋅( −1 )⋅( −1 ) =−5

Example 2:

Simplify.

( 2i )( −6i )( −7i )

Rewrite the expression grouping the real and imaginary parts.

( 2i )( −6i )( −7i )=( 2 )⋅( −6 )⋅( −7 )⋅i⋅i⋅i =84⋅i⋅i⋅i

Simplify the imaginary part using the property of multiplying powers.

=84 i 3 =84 i 2 i 1

Recall the definition of i .

Since i 2 =−1 :

=84( −1 )i =−84i

Example 3:

Simplify the principal square roots.

−64

Taking the square root and substituting −1 =i :

= −1 ⋅ 64 =i 64

Simplify.

=8i

Example 4:

Simplify the principal square roots.

−121 x 4

Factor the radicand into squares,

−121 x 4 = −1⋅121⋅ x 4 =11 x 2 −1

Rewrite the expression using i .

=11 x 2 i

Note: When you are simplifying radicals as part of an equation, please remember that unless a principal square root is requested, there are always both positive and negative roots - including when you are working with i .