Question

Find the square root of the following complex numbers.
i) $1+4\sqrt{3i}$
ii) $3+2\sqrt{10}i$
iii) $2(1-\sqrt{3}i)$

Answer 1

We can solve this problem using the folloing:

$(a\pm bi{)}^2=(a^2-b^2)\pm 2abi$

$\therefore \sqrt{(a\pm bi{)}^2}=a\pm bi=\sqrt{(a^2-b^2)\pm 2abi}$

(i)

$1+4\sqrt{3}i$

$=1.99+i1.73$

$\sqrt{1+4\sqrt{3}i}=1.99+i1.73$

(ii)

$3+2\sqrt{10}i$

$=2.23+i1.41$

$\sqrt{3+2\sqrt{10}i}=2.23+i1.41$

(iii)

$2(1-\sqrt{3}i)$

$2-2\sqrt{3}i$

$=1.729-i0.998$

$\sqrt{2(1-\sqrt{3}i)}=1.729-i0.998$