```Evaluating Powers of
1 =
2 = −1
See the pattern:
3 =  2 ∗  1 = −1 ∗  = −

4 =  2 ∗  2 = −1 ∗ −1 = 1
−1
5 =  2 ∗  2 ∗  1 = −1 ∗ −1 ∗  =
−
6 =  2 ∗  2 ∗  2 = −1 ∗ −1 ∗ −1 = −1
1
7 =  2 ∗  2 ∗  1 = −1 ∗ −1 ∗  = −
8 =  2 ∗  2 ∗  2 ∗  2 = −1 ∗ −1 ∗ −1 ∗ −1 = 1
Simplify.
42 = ( 2 )21 = (−1)21 = −1
21 = ( 2 )10 ∗  = (−1)10 ∗  = 1 ∗  =
83 = ( 2 )41 ∗  = (−1)41 ∗  = −1 ∗  = −
Imaginary Numbers
√−1 =
2
(√−1) =  2
−1 =  2
Simplify.
√−16 = √16 ∗ √−1 = 4
√−20 = √20 ∗ √−1 = 2√5 ∗  = 2√5
(4)2 = 4 ∗ 4 = 16 2 = 16 ∗ −1 = −16
√−9 ∗ √−25 = 3 ∗ 5 = 15 2 = 15 ∗ −1 = −15
Always factor out a √−1 , which is equal
to  , then simplify.
(−3 + 5) + (−6 − 8)
For subtraction do not forget to
change every sign past the minus sign
between sets of parenthesis.
Then, just combine like terms. Real
with real, and Imaginary with
imaginary.
(2 + 3) − (3 + 5)
Multiplying Complex Numbers
2(3 − 5)
Distribute.
FOIL (first, outer, inner, last)
Don’t forget:  2 = −1
(3 + 2)(3 − 4)
Dividing Complex Numbers
3−
2−
3+8
−
Multiply by the
conjugate: (a + bi)
Multiply by the
conjugate: (+i)
Conjugate: a binomial formed by making the second term of a binomial negative.
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