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Need some hints on how to Solve $\sqrt {35 - 5i}$

Attempt.

I factorized 5 out and it became $\sqrt {5(7-i)}$

I just want to know if it can be solved further.

Thanks.

Antonio Vargas
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john scott
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    When you are solving something, you are trying to find the value of something which is unknown. As it stands, you don't have any unknowns in your expression. Can you be clearer about what you are trying to do? – Bernard W Jan 28 '16 at 06:03
  • You can compute the square root by first expressing the complex number in exponential form. – Deepak Jan 28 '16 at 06:08

2 Answers2

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You want to solve $$(x+iy)^2=35-5i\ .$$ Working out the square and equating real and imaginary parts, $$x^2-y^2=35\ ,\quad 2xy=-5\ .$$ Multiply the first by $4x^2$ to get $$4x^4-4x^2y^2=140x^2$$ and substitute from the other equation, $$4x^4-25=140x^2\ .$$ You can now solve this as a quadratic in $u=x^2$. One of the values will have to be rejected, you then get two possible values for $x$ and two corresponding values for $y$.

David
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  • Thanks a lot Mr. David. I got x as ±5.9 and y as ±5.37. Is this correct sir? – john scott Jan 28 '16 at 07:18
  • Note that $xy$ is negative. So if $x$ is positive then $y$ must be negative, and if $x$ is negative then $y$ must be positive. I do not have a calculator with me but you can check your answer by calculating $(x+iy)^2$. – David Jan 28 '16 at 09:23
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One way is to rewrite the $35-5i$ on polar form. We have that $|35-5i| = \sqrt{35^2+5^2} = \sqrt{1250} = 25\sqrt2$. The argument you get as $\varphi=\arctan{-5/35} = -\arctan{1/7}$. So we have:

$$\sqrt{35-5i} = \sqrt{25\sqrt2e^{-i\arctan{1/7}}} = 5\sqrt{\sqrt{2}}e^{-{i\over2}\arctan{1/7}+in\pi}$$

You could also approach it by setting solving $(x+iy)^2 = 35-5i$ and identifying real and imaginary parts. This leads to a biquadratic equation, whose expression is also quite complex.

I think that the expression $\sqrt{5(7-i)}$ is as simple as it gets, but there may be reasons that you actually want it in explicit cartesian or polar form that would motivate a more complex expression.

skyking
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