Express the following complex numbers in `a+i b` form: `((3-2i)(2+3i))/((1+2i)(2-i))` (ii) `(2-sqrt(-25))/(1-sqrt(-16))`

Answer

(a) `(63)/(25) -(16)/(25)i`
(b) `(12)/(17) + (3)/(17)i`

Explanation

(a) `((3-2i)(2+3i))/((1+2i)(2-i)) = ((6+6)+i(-4+9))/((2+2)+i(4-1))`
`- (12+5i)/(4+3i)`
` (12 + 5i)/(4+3i)(4-3i)/(4-3i)`
`= ((48 = 15)+i(-36+20))/(16-9i^(2))`
`= (63)/(25)-(16)/(25i)`
(b) `(2-sqrt(25))/(1-sqrt(-16)) = (2-5i)/(1-4i)`
`= (2-5i)/(1-4i)xx(1+4i)/(1+4i)`
`= ((2+20)+i(8-5))/(1-16i^(2))`
`= (22+ 3i)/(17) = (22)/(17) + (3)/(17) i`
(c) `((2i)/( 1+i))^(2) = ((2i(1-i))/((1+i)(1-i))^(n)`
` = ((2(i-i)^(2))/(2))^(n)`
`= (i+1)^(n)`
` = (2i)^(n//2)`
Hence n = 8 is the least positive integer for which the given complex number is a positive integer.