Question

square root of 20-21i

Answer 1

4.47213595 - 21 i
.....
@@ @@ @@ @@ @@

Answer 2

Let 20 - 21i = ( a + bi )² for (a,b)∈R
=> 20 - 21i = a² + (2ab)i + (b²)i²
=> 20 - 21i = (a² - b²) + (2ab)i

Equating equations:
a² - b² = 20
2ab = -21

=> b= -21/2a

Putting this in previous eqn.,
a² - 441/(4a²) = 20
=> 4a². a² - 80 a² - 441 = 0
=> 4y² - 80y -441 = 0
=> y = [ ( 80 ± √(6400 + 441 * 16) ) /8 ]
=> y = [ (80 ± √13456) / 8 ]
=> y = [ ( 80 ± 116 ) / 8 ]
=> y = [10 ± 14.5 ] 
=> y = 24.5                   ['.' a∈R ]
=>a² = 24.5
=> a = ± √24.5 = ±4.95
=> b² = a² - 20 = 4.5
=> b = ± √4.5 = ± 2.12

=> 20 - 21i = (4.95 - 2.12i)²
=>√(20 - 21i) ≈ ±(4.95 - 2.12i)