Question

Find the square root of 2-2√3i

Answer 1

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Answer 2

Answer:

Roots are $\pm(\sqrt{3}-i)$

Step-by-step explanation:

To find : The square root of $2-2\sqrt{3}i$

Solution:

Consider, $z=x+iy=2-2\sqrt{3}i$

x=2 and $y=-2\sqrt{3}$

Now, $|z|=\sqrt{x^2+y^2}$

$|z|=\sqrt{2^2+(-2\sqrt{3})^2}$

$|z|=\sqrt{4+12}$

$|z|=\sqrt{16}$

$|z|=4$

Here y<0, so square root of $2-2\sqrt{3}i$ are

$R=\pm(\sqrt{\frac{|z|+x}{2}}-i\sqrt{\frac{|z|-x}{2}})$

$R=\pm(\sqrt{\frac{4+2}{2}}-i\sqrt{\frac{4-2}{2}})$

$R=\pm(\sqrt{\frac{6}{2}}-i\sqrt{\frac{2}{2}})$

$R=\pm(\sqrt{3}-i\sqrt{1})$

$R=\pm(\sqrt{3}-i)$