Simplify √(5-21i) and write the answer in the form `a+bi, where a,b ∈ R

Answer 1

let x = 5 - 21i
|x| = √(25 + 441) = √466
tanθ = -21/5
θ = 360° - 76.608° = 283.392° , since 5 - 21i is in quad IV on the Argand plane

x^(1/2) = √√466(cos (1/2)283.392°+ i sin (1/2)283.392) , using De Moivre
= √√466(cos 141.696 + i sin 141.696)

= 4.646(-.784735.. + i .61983..)
= -.3646 + 2.8798 i <------ primary square root

other root = √√466(cos (141.696 + 180°) + i sin (141.696 + 180°) )
leaving that calculation up to you

checking
( 3.646 + 2.8798 i)^2
= 13.2935 - 21i + 8.2935 i^2
= 13.2935 - 21i - 8.2935
= 5 - 21 i