Simplify and express each of the following in the form (a + ib) :
`{:((i),(2 + sqrt(-3))^(2),(ii),(5-2i)^(2),(iii),(-3+5i)^(3)),((iv),(-2-(1)/(3)i)^(3),(v),(4-3i)^(-1),(vi),(-2+sqrt(-3))^(-1)),((vii),(2+i)^(-2),(viii),(1+2i)^(-3),(ix),(1+i)^(3)-(1-i)^(3)):}`

Answer

`{:((i),(1+4 sqrt(3)i),(ii),(21-20i),(iii),(198+10i),"(iv)",((-22)/(3)-(107)/(27)i)),((v),((4)/(25)+(3)/(25)i),(vi),((-2)/(7)-(sqrt(3))/(7)i),(vii),((3)/(25)-(4)/(25)i),"(vii)",((-11)/(125)+(2)/(125)i)),((ix),4i,,,,,,):}`

Explanation

`(iii) (-3+5i)^(3) = (5i-3)^(3) = (5i)^(3)-3^(3) - (3xx5ixx3)(5i-3)=(-125i-27+225+135i)=(198+10i)`.
`(iv) (-2-(1)/(3)i)^(3)=(-1)^(3)(2+(1)/(3)i)^(3) = -{8+(1)/(27)i^(3)+2i(2+(!)/(3)i)}`.
`(v) (4-3i)^(-1)=(1)/((4-3i))xx((4+3i))/((4+3i))=((4+3i))/(25)`.
`(vi) (-2+sqrt(3)i)^(-1) = (1)/((-2+sqrt(3)i))xx((-2-sqrt(3)i))/((-2+sqrt(3)i))`.
`(vii)(2+i)^(-2)=(1)/((2+i)^(2))=(1)/((4+i^(2)+4i))=(1)/((3+4i))xx((3-4i))/((3-4i))`.
`(viii) (1 + 2i)^(-3)=(1)/((1+2i)^(3))=(1)/({1+8i^(3)+6i(1+2i)})=(1)/((-11+2i))xx((-11+2i))/((-11+2i))`.