Question

Find the square root of 1+i​

Answer 1

Step-by-step explanation:

We have,

$z = 1 + i$

$z = \sqrt{2} ( \frac{1}{ \sqrt{2} } + \frac{i}{ \sqrt{2} })$

$\implies \: z = \sqrt{2} (e^{ \frac{i\pi}{4} } )$

$\implies \sqrt{z} = {2}^{ \frac{1}{4} } . {e}^{i \frac{\pi}{8} }$

$= {2}^{ \frac{1}{4} } .( \cos( \frac{\pi}{8} ) + i \sin( \frac{\pi}{8} ) )$

$= ±{2}^{ \frac{1}{4} } .( \sqrt{ \frac{ \sqrt{2} + 1 }{2 \sqrt{2} } } + i \sqrt{ \frac{ \sqrt{2} - 1 }{2 \sqrt{2} } } )$

$= ± \frac{1}{ \sqrt{2} } ( \sqrt{ \sqrt{2} + 1} + i \sqrt{ \sqrt{2} - 1} )$