Question

find the square root of 6+8i

Answer 1

(6+8i)²= (6)²+2(6)(8i)+(8i)²

=36+96i+64(-1)

=96i+36-64

=96i-28

Answer 2

Let √( 6 + 8 i ) = a + i b

Squaring both sides ,

6 + 8 i = a² - b² + 2ab i { As i² = -1}

Equating real and imaginary parts,

a² - b² = 6.......(1) , 2 ab = 8 ......(2)

We have identity

[ a² + b² ] ² = [ a² - b² ]² + (2ab)²

[ a² + b² ] ² = [6]² + (8)²

[ a ² + b ² ] ² = 36 + 64

a² + b ² = 10 .........(4)

adding (1) and (4)

2 a ² = 16

a² = 8 => a= ± 2√2

put a in (1)

b² = a ²- 6

b² = 8 - 6

b = ± √ 2

Now as 2ab is positive so a and b are of same sign

Hence ,, √ ( 6 + 8 i) = ± ( 2√2 + √2 i)