Let the required square root of -7 + 24i be x + yi

x + yi = sqrt ( -7 + 24i)

square both the sides

x^2 - y^2 + 2xyi = -7 + 24i

equate the real and complex coefficients

x^2 - y^2 = -7

2xy = 24

=> xy = 12

=> x = 12/y

Substitute in x^2 - y^2 = -7

=> 12^2/y^2 - y^2 = -7

=> 144 - y^4 + 7y^2 = 0

=> y^4 - 7y^2 - 144 = 0

let u = y^2

=> u^2 - 7u - 144 = 0

=> u^2 - 16u + 9u - 144 = 0

=> u(u - 16) + 9(u - 16) = 0

=> (u - 16)(u + 9) = 0

=> u = 16 and u = -9

y = sqrt u is a real number so we take only u = 16.

y^2 = 16 , y = 4 and y = -4

x = 12/y = 3 and -3

**The required value of sqrt( - 7 + 24i) = -3 - 4i and 3 + 4i**