roots

**meli3000** · April 14th 2008, 12:37 PM

Explain how you would find the square roots of 5+3i then use the method to find the roots of 5+3i

Please show all steps!

**Peritus** · April 14th 2008, 12:49 PM

there are 2 ways, one is long and one is short,

The short one:

$ 5 + 3i = \sqrt {34} e^{j\arctan \left( {\frac{3} {5}} \right)} $

$ \Rightarrow \sqrt {5 + 3i} = \sqrt[4]{{34}}e^{j\frac{1} {2}\arctan \left( {\frac{3} {5}} \right)} = \sqrt[4]{{34}}\left[ {\cos \left( {\frac{1} {2}\arctan \left( {\frac{3} {5}} \right)} \right) + j\sin \left( {\frac{1} {2}\arctan \left( {\frac{3} {5}} \right)} \right)} \right]$

and of course the second root:

$ \sqrt[4]{{34}}e^{j\frac{{\arctan \left( {\frac{3} {5}} \right) + 2\pi }} {2}} $

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