# Find the modulus and argument of sqrt(3+i).

#### ach4124

I tried letting sqrt(3+i) = a+bi. With some calculations, I got 2a^2 = x + sqrt(x^2+y^2), 2b^2 = -x + sqrt(x^2+y^2), which means a = +- sqrt[(3+sqrt(10))/2] and b = +- sqrt[(-3+sqrt(10))/2]. Does this mean there are four possible answers?

But then when I tried doing this by expressing the complex number in polar coordinate form, there are only two roots.

sqrt(3+i) = 10^(1/4) [cos (1/2)tan-1(1/3) + i sin (1/2)tan-1(1/3)]

or 10^(1/4) {cos [(1/2)tan-1(1/3)+pi] + i sin [(1/2)tan-1(1/3)+pi]}

What's wrong?

#### mathman

Forum Staff
ab = 1/2, so the allowable solutions in the first approach are those where a and b have the same sign.

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