Solution:
If ![b>0, \sqrt{a+i b}=\pm\left[\left(\frac{a+\sqrt{a^{2}+b^{2}}}{2}\right)^{\frac{1}{2}}+i\left(\frac{-a+\sqrt{a^{2}+b^{2}}}{2}\right)^{\frac{1}{2}}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-5ffa871c6cd7675f669d73da69f0953d_l3.png "Rendered by QuickLaTeX.com")
If ![b<0, \sqrt{a+i b}=\pm\left[\left(\frac{a+\sqrt{a^{2}+b^{2}}}{2}\right)^{\frac{1}{2}}-i\left(\frac{-a+\sqrt{a^{2}+b^{2}}}{2}\right)^{\frac{1}{2}}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-bc2b1e8b97142f16b93977b37fecd267_l3.png "Rendered by QuickLaTeX.com")
(i) 
It is known that, 
Therefore, 
Here, 
Let’s simplify now,
![\begin{aligned} \sqrt{-5+12 i} =\pm\left[\left(\frac{-5+\sqrt{(-5)^{2}+12^{2}}}{2}\right)^{\frac{1}{2}}+i\left(\frac{5+\sqrt{(-5)^{2}+12^{2}}}{2}\right)^{\frac{1}{2}}\right][\text { Since, } b>0] \\ =\pm\left[\left(\frac{-5+\sqrt{25+144}}{2}\right)^{\frac{1}{2}}+i\left(\frac{5+\sqrt{25+144}}{2}\right)^{\frac{1}{2}}\right] \\ =\pm\left[\left(\frac{-5+\sqrt{169}}{2}\right)^{\frac{1}{2}}+i\left(\frac{5+\sqrt{169}}{2}\right)^{\frac{1}{2}}\right] \\ =\pm\left[\left(\frac{-5+13}{2}\right)^{\frac{1}{2}}+i\left(\frac{5+13}{2}\right)^{\frac{1}{2}}\right] \\ =\pm\left[\left(\frac{8}{2}\right)^{\frac{1}{2}}+i\left(\frac{18}{2}\right)^{\frac{1}{2}}\right] \\ =\pm\left[4^{\frac{1}{2}}+i 9^{\frac{1}{2}}\right] \\ =\pm[2+3 i] \end{aligned}](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-04175a027b1bd527e1b1c89e10e68bf7_l3.png "Rendered by QuickLaTeX.com")
As a result, square root of 
![\pm[2+3 i]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-7b8c04091b27029b6361877b063164f5_l3.png "Rendered by QuickLaTeX.com")
(ii) 
It is known that, 
Therefore, 
Here, 
Let’s simplify now,
![\begin{aligned} \sqrt{-7-24 i} =\pm\left[\left(\frac{-7+\sqrt{(-7)^{2}+(-24)^{2}}}{2}\right)^{\frac{1}{2}}-i\left(\frac{7+\sqrt{(-7)^{2}+(-24)^{2}}}{2}\right)^{\frac{1}{2}}\right]_{[\text {Since, } b<0]} \\ =\pm\left[\left(\frac{-7+\sqrt{49+576}}{2}\right)^{\frac{1}{2}}-i\left(\frac{7+\sqrt{49+576}}{2}\right)^{\frac{1}{2}}\right] \\ =\pm\left[\left(\frac{-7+\sqrt{625}}{2}\right)^{\frac{1}{2}}-i\left(\frac{7+\sqrt{625}}{2}\right)^{\frac{1}{2}}\right] \\ =\pm\left[\left(\frac{-7+25}{2}\right)^{\frac{1}{2}}-i\left(\frac{7+25}{2}\right)^{\frac{1}{2}}\right] \\ =\pm\left[\left(\frac{18}{2}\right)^{\frac{1}{2}}-i\left(\frac{32}{2}\right)^{\frac{1}{2}}\right] \\ =\pm\left[9^{1 / 2}-\mathrm{i} 16^{1 / 2}\right] \\ =\pm[3-4 \mathrm{i}] \end{aligned}](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-8da78174af30876eb2ad90123b9ca76d_l3.png "Rendered by QuickLaTeX.com")
As a result, square root of 
![\pm[3-4 i]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-9586c871ab1c0f1dee276e404784490e_l3.png "Rendered by QuickLaTeX.com")
Solution:
If
If
(i)
It is known that,
Therefore,
Here,
Let’s simplify now,
As a result, square root of
(ii)
It is known that,
Therefore,
Here,
Let’s simplify now,
As a result, square root of
