Question

find the square root of 3 + 4i

Answer 1

here is your answer....

Answer 2

$\large{\bold{ \underline{ \underline { \: \: Given \: \: }}}}$

$\to$Complex number [Z] = 3 + 4i

$\to$Imaginary part [Im(Z)] = 4

$\to$Real part [Re(Z)] = 3

$\large{\bold{ \underline{ \underline { \: \: To \: Find \: \: }}}}$

$\to \bold{ \sqrt{z} = \: ? }$

$\large{\bold{ \underline{ \underline { \: \: Solution \: \: }}}}$

Imaginary part > 0 , Therefore ,

$\fbox{ \fbox{ \bold{ \: \: \sqrt{z} = ± \: ( \: \: \sqrt{ \frac{ |z| + Re(Z) }{2} } + \sqrt{ \frac{ |z| - Re(Z)}{2} } \: \: ) \: \: \: }}}$

$\to \bold{ \sqrt{z} = ± \: ( \sqrt{ \frac{ 5 + 3 }{2} } +i \sqrt{ \frac{ 5 - 3}{2} } ) }\\ \\ \to \bold{\sqrt{z} = ± \: ( \sqrt{ \frac{ 8}{2} } +i \sqrt{ \frac{ 2}{2} } )} \\ \\\to \bold{ \sqrt{z} = ± \: ( \sqrt{ 4 } +i \sqrt{ 1}) } \\ \\\to \bold{ \sqrt{z} = ± \: ( 2 +i 1) }$

Hence , $\bold{± \: ( 2 +i 1) }$ is the square root of given complex number