Find the square root `9+40 idot`

Answer

`(5+4i) or - (5+4i)`

Explanation

Let `sqrt(9+40i) = x +iy`. Then
`(x + iy)^(2) = 9+40i`
`or x^(2) - y^(2) = 9" "(1)`
`and xy = 20" "(2)`
Squaring (1) and adding with 4 times the square of (2), we get
`x^(4) + y^(4) - 2x^(2)y^(2)+ 4x^(2)y^(2) = 81 + 1600`
`or (x^(2) +y^(2))= 1681`
or `x^(2) + y^(2) = 41" "(3)`
From`(1) +(3)`, we get
`x^(2)= 25 or x = pm 5` and `y =pm4`
From Eq. (2) we can see that x and y are of same sign. Therefore,
`x+iy= (5+4i) or - (5+41)`