Ex 5.1, 10 - Express in a + ib: (-2 - 1/3i)3 - Chapter 5 - Identities (square, cube of 2 complex numbers)

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Ex5.1, 10 Express the given Complex number in the form a + ib: (โˆ’2โˆ’1/3 ๐‘–)^3 (โˆ’2โˆ’1/3 ๐‘–)^3 = โˆ’ 1 (2+1/3 ๐‘–)^3 = โˆ’ (2+1/3 ๐‘–)^3 It is of the form ( a + b )3 Using ( a + b )3 = a3 + b3 + 3 ab ( a + b ) Here a = 2 and b = 1/3 i = โˆ’((2)^3+(1/3 ๐‘–)^3+3 ร—2ร—1/3 ๐‘–(2+1/3 ๐‘–)) = โˆ’(8+(1/3)^2ร—(๐‘–)^3+2๐‘–(2+1/3 ๐‘–)) = โˆ’(8+1/27 ๐‘–^3+4๐‘–+2/3 ๐‘–^2 ) = โˆ’(8+1/27 ใ€–๐‘–ร—๐‘–ใ€—^2+4๐‘–+2/3 ๐‘–^2 ) = โˆ’1 (8+1/27 ๐‘–(โˆ’1)+4๐‘–+2/3 (โˆ’1)) = โˆ’1 (8+1/27 ๐‘–+4๐‘–โˆ’2/3) = โˆ’ 8 + 1/27 ๐‘– + 4 i + 2/3 = โˆ’ 8 + 2/3 + ( 1)/27 ๐‘– โ€“ 4 ๐‘– = (โˆ’ 8+ 2/3) + (( 1)/27โˆ’4)๐‘– = ((โˆ’ 24 +2)/3) + ((1 โˆ’ 4 ร—27)/27)๐‘– = (โˆ’ 22)/3 + ((1 โˆ’108)/27)๐‘– = (โˆ’ 22)/3 + (( โˆ’107)/27)๐‘–

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