Use the method in the Example to find the square roots of 1 + √3i

Question

asked Feb 28, 2020 152k views

1 Answer

Irwinb · Answer 1 · 2020-03-04T20:17:33+0000

Answer:

$\sqrt{1 + √(3) i}=\pm (1.58+i 0.548)$

Explanation:

Given

z = 1 + √3 i

Let
$√(1+\sqrt(3) i)=p+iq$

Squaring both sides

$1+\sqrt(3) i=p^2-q^2+2ipq$

Comparing real and imaginary part

Re(LHS)=Re(RHS)

1=p^2-q^2 ...........................(1)

comparing Im(LHS)=Im(RHS)

√3=2pq

q=(√(3))/(2p)

Substitute q in equation (1)

1=p^2-((√(3))/(2p))^2

p^4-p^2-0.75=0

Let
x=p^2

x^2-x-0.75=0

$x=(1\pm √(1^2+4* 0.75))/(2)$

$x=(1\pm 4)/(2)$

we take only Positive value because
p^2=x

x=2.5

p^2=2.5

thus
$p=\pm 1.58$

$q=\pm 0.548$

thus,

$\sqrt{1 + √(3) i}=\pm (1.58+i 0.548)$