Question

$[/tex] [tex] \sqrt{9 + 40i} + \sqrt{9 +40i}$

Answer 1

2 $\sqrt{9 + 40i}$ 

Now,
9 + 40i = (a+bi)² = (a² - b²) + 2abi

Equating thy equation: 
=> (a² - b²) = 9 ; 2ab = 40
=> ab = 20
=> b = 20/a => (a² - (20/a)²) = 9
Let a²  =  y;
Now,

y² - 9y - 400 = 0
=> 2y = 9 + √(81 + 1600) ; 9 - √(81 + 1600)
=> 2y = 9 + √1681 ; 9 - √1681
=> 2y = 9 + 41 ; 9 - 41
=> 2y = 50 ; -32
=> y = 25; -16
Now, since (a,b)∈R 
=> a² = 25
=>a = 5 or -5
Now,
Accordingly,
b = 4 or -4

=> ( 9 + 40i ) = ( 5 + 4i )²
=> 2 $\sqrt{9 + 40i}$ = 2(5 + 4i) = 10 + 8i or -10 - 8i