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. Adding and Subtracting Complex Numbers (−3 + 5) + (−6 − 8) Nothing to distribute for addition. 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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# Complex and Imaginary Numbers