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hypothesis

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**1. The complex number (C) is [tex] C = 1/\sqrt{i*x} [/tex]. find the two roots of C. The solution for one of the roots is given as [tex] C = \sqrt{1/2x} - i\sqrt{1/2x} [/tex] can someone show me how to get to that solution?**

**2. Equations: the relationships between polar and rectangular coordinates for complex variables**

**3. I try taking the square root of i*x by using the relation:**

[tex] \sqrt{a+ib} = \sqrt{r}\left(cos(\theta/2)+i*sin(\theta/2)) [/tex]

Then substitute with my variables (a+ib) = (0+ix) and because the real part is zero and the imaginary part is positive the angle is [tex] \pi/2 [/tex] so I get:

[tex]

\sqrt{0+ix} = \sqrt{\sqrt{0^2+x^2}}(cos(\pi/4)+i*sin(\pi/4))

[/tex]

=

[tex]

\sqrt{ix} = \sqrt{x}(cos(\pi/4)+i*sin(\pi/4)

[/tex]

Substitute it back into the first equation ([tex] C = 1/\sqrt{i*x} [/tex]) and get:

[tex] C = \frac{1}{\sqrt{x}(cos(\pi/4)+i*sin(\pi/4)} [/tex]

now I can multiply nominator and denominator with the conjugate of the denominator and come up with a solution:

[tex] C = \frac{\sqrt{x}}{2x} - i \frac{1}{2x} [/tex]

but this is not the solution that I have been given plus I was supposed to find two solutions so I'm clearly doing something wrong. But what?

[tex] \sqrt{a+ib} = \sqrt{r}\left(cos(\theta/2)+i*sin(\theta/2)) [/tex]

Then substitute with my variables (a+ib) = (0+ix) and because the real part is zero and the imaginary part is positive the angle is [tex] \pi/2 [/tex] so I get:

[tex]

\sqrt{0+ix} = \sqrt{\sqrt{0^2+x^2}}(cos(\pi/4)+i*sin(\pi/4))

[/tex]

=

[tex]

\sqrt{ix} = \sqrt{x}(cos(\pi/4)+i*sin(\pi/4)

[/tex]

Substitute it back into the first equation ([tex] C = 1/\sqrt{i*x} [/tex]) and get:

[tex] C = \frac{1}{\sqrt{x}(cos(\pi/4)+i*sin(\pi/4)} [/tex]

now I can multiply nominator and denominator with the conjugate of the denominator and come up with a solution:

[tex] C = \frac{\sqrt{x}}{2x} - i \frac{1}{2x} [/tex]

but this is not the solution that I have been given plus I was supposed to find two solutions so I'm clearly doing something wrong. But what?